GIFT  OF 
President's   Office 


THREE-COLOR    PRINTING 
(See  pane  236) 


FIRST   PRINCIPLES 


OF 


PHYSICS 


BY 


HENRY   S.    CARHART,  LL.D. 
\> 

PROFESSOR   EMERITUS   OF   PHYSICS,    UNIVERSITY   OF   MICHIGAN 
AND 

HORATIO  N.   CHUTE,    M.S. 

INSTRUCTOR  IN  PHYSICS  IN  THE  ANN  ARBOR  HIGH  SCHOOL 


Boston 

ALLYN    AND    BACON 
1912 


COPYRIGHT,  1912 
BY  HENRY  S.  CARHART 
AND  HORATIO  N.  CHUTE 


J.  S.  Cashing  Co.  —  Berwick  &  Smith  Co. 
Norwood,  Mass.,  U.S.A. 


PREFACE 

NOT  many  years  ago  the  ideal  textbook  in  elementary  physics 
was  little  more  than  a  concise,  scientific  statement  of  the  fun- 
damental principles  of  the  subject,  illustrated  by  experiments 
and  reenforced  by  numerous  problems  designed  to  emphasize 
the  expression  of  physical  relations  in  mathematical  form. 
Teachers,  however,  came  to  recognize  that  books  of  this  type 
do  not  make  the  subject  sufficiently  attractive  to  pupils  who 
are  familiar  with  picturesque,  popular  applications  of  physics, 
and  who  are  not  inclined  to  devote  the  requisite  time  and  effort 
to  overcome  the  difficulties  of  a  less  alluring  side  of  the  subject. 

Attempts  have  been  made  to  meet  this  vague  dissatisfaction 
by  presenting  many  familiar  illustrations  of  the  practical  appli- 
cations of  physics,  but  with  an  unsatisfactory  treatment  of  the 
fundamental  principles  on  which  these  applications  are  based. 
As  a  result,  pupils  reached  the  end  of  their  study  with  few 
definite  ideas  and  little  knowledge  of  the  science  itself.  This 
condition  was  plainly  so  much  worse  than  the  former  one  that 
teachers,  who  have  been  drawn  into  the  experiment,  have  gen- 
erally preferred  to  return  to  the  earlier  type  of  book. 

The  present  volume  is  the  result  of  an  attempt  to  make  a 
book  which  shall  have  a  strong  element  of  interest  and  attrac- 
tiveness, and  at  the  same  time  be  so  clear  and  definite  in  the 
treatment  of  principles  that  pupils  may  carry  away  from  the 
course  some  useful  acquisitions  for  daily  life  and  a  preparation 
for  continuing  the  subject  with  success  in  the  college  or  tech- 
nical school.  Every  effort  has  been  made  to  have  the  language 
of  the  book  clear  and  simple.  Cordial  acknowledgment  is  made 
to  many  teachers  whose  helpful  criticisms  have  been  an  aid  to 

239420 


IV  PREFACE 

this  end.  The  problems  given  require  for  their  solution  only 
such  simple  algebraic  operations  as  offer  no  difficulty  to  pupils 
who  have  taken  a  course  in  elementary  algebra. 

Yielding  to  the  advice  of  teachers  in  whose  judgment  the  au- 
thors have  the  greatest  confidence,  a  somewhat  radical  change 
has  been  made  in  the  order  of  subjects  by  placing  the  Mechanics 
of  Fluids  before  the  Mechanics  of  Solids.  The  reason  assigned 
is  that  the  facts  and  principles  of  the  former  are  more  familiar 
than  those  of  the  latter ;  they  thus  form  a  natural  approach 
to  the  abstract  laws  of  motion,  the  composition  and  resolution 
of  forces  and  velocities,  and  the  formulae  connected  with  them 
as  the  shorthand  expression  of  physical  laws. 

H.  S.  C. 

H.  N.  C. 
April,  1912. 


CONTENTS 

Chapter  I.    Introduction  PAGE 

I.     Matter  and  Energy 1 

II.     Properties  of  Matter     .         .        .        .        .         .        .        3 

III.     Physical  Measurements  -    .         .        .        .        .        .10 

Chapter  II.     Molecular  Physics 

I.     Molecular  Motion         .  .        .        .        .         .16 

II.     Surface  Phenomena .19 

III.     Molecular  Forces  in  Solids 25 

Chapter  HI.     Mechanics  of  Fluids 

I.  Pressure  of  Fluids 31 

II.  Bodies  Immersed  in  Liquids 40 

III.  Density  and  Specific  Gravity 44 

i          IV.  Pressure  of  the  Atmosphere 49 

V.  Compression  and  Expansion  of  Gases          ...       57 

VI.  Pneumatic  Appliances          .  .         .         .         .66 

'"Chapter  IV.     Motion 

I.     Motion  in  Straight  Lines 71 

II.     Curvilinear  Motion       .......       79 

III.     Simple  Harmonic  Motion    ......       80 

Chapter  V.     Mechanics  of  Solids 

I.  Measurement  of  Force          ......       83 

II.  Composition  of  Forces  and  of  Velocities     ...       86 

III.  Newton's  Laws  of  Motion    ......       93 

IV.  Gravitation .         .98 

V.     Falling  Bodies 104 

VI.     Centripetal  and  Centrifugal  Force       .         .         .         .109 
VII.     The  Pendulum  .         .  Ill 


vi  CONTENTS 


Chapter  VI.     Mechanical  work 

PAGE 

I.     Work  and  Energy       .... 
II.     Machines    

.    117 
.     126 

Chapter  VII.     Sound 

I.     Wave  Motion      

.        .        .143 

II.     Sound  and  its  Transmission 

.     148 

III.     Velocity  of  Sound       .... 
IV.     Reflection  of  Sound    . 

.     150 
.     153 

V.     Resonance  ...... 

.     155 

VI.     Characteristics  of  Musical  Sounds 

.     157 

VII.     Interference  and  Beats 

.     160 

VIII.     Musical  Scales     

.     162 

IX.     Vibration  of  Strings   .... 
X.     Vibration  of  Air  in  Pipes  . 

.     167 
.     171 
.     174 

Chapter  VIII.     Light 

I.     Nature  and  Transmission  of  Light    . 
II.     Photometry          
III.     Reflection  of  Light      .... 
IV.     Refraction  of  Light     .... 
V.     Lenses          ...... 

.     179 

.     184 
.     187 
.     203 
.     210 

VI.     Optical  Instruments    .... 

.     219 
.     296 

VIII.     Color  

.     233 

.     238 

Chapter  IX.     Heat 

I.     Heat  and  Temperature       .        .        .    . 
II.     The  Thermometer      .... 

.     242 
.     244 

III.     Expansion  ...... 
IV.     Measurement  of  Hea«t 

.     248 
.    257 

V.     Change  of  State  
VI.     Transmission  of  Heat 

.     260 
.     267 

VII.    Heat  and  Work. 

274 

CONTENTS  Vll 


Chapter  X.     Magnetism 

PAOK 

I.     Magnets  and  Magnetic  Action  . 

.     281 

II.     Nature  of  Magnetism 

.     286 

III.     The  Magnetic  Field   .... 

.     287 

IV.    Terrestrial  Magnetism 

.     290 

^Chapter  XI.    Electrostatics 

I.     Electrification     

.    293 

II.     Electrostatic  Induction 

.    298 

III.     Electrical  Distribution 

.    300 

IV.     Electric  Potential  and  Capacity 

.     302 

V.     Electrical  Machines    .... 

.     308 

VI.     Atmospheric  Electricity     . 

.     313 

Chapter  XII.    Electric  Currents 

I.     Voltaic  Cells       

.    316 

II.     Electrolysis          ..... 

.     327 

III.     Ohm's  Law  and  its  Applications 

.    332 

IV.     Heating  Effects  of  a  Current      . 

.     338 

V.     Magnetic  Properties  of  a  Current 

.        .        .340 

VI.     Electromagnets  

.     344 

VII.     Measuring  Instruments 

.     347 

Chapter  XIII.    Electromagnetic  Induction 

I.     Faraday's  Discoveries 

.     354 

II.     Self-induction     ..... 

.     358 

III.     The  Induction  Coil     .         . 

.     359 

Chapter  XIV.    Dynamo-Electric  Machinery 

I.     Direct  Current  Machines    . 

.     371 

II.     Alternators  and  Transformers   . 

.     378 

III.     Electric  Lighting        ... 

.  .381 

IV.     The  Electric  Telegraph      . 

.     384 

*    V.     The  Telephone   

.     388 

VI.     Wireless  Telegraphy  .... 

.     390 

viii  CONTENTS 

Appendix  PAGE 

I.  Geometrical  Constructions 394 

II.  Conversion  Tables      .        .         .         .         .         .         .  398 

III.  Mensuration  Rules 400 

IV.  Table  of  Densities .401 

V.  Geometrical  Construction  for  Refraction  of  Light    .  402 

Index     ...  405 


PORTRAITS   OF   EMINENT   PHYSICISTS 

FACING   PAGE 

Blaise  Pascal 32 

Galileo  Galilei .         .        .  •      .  32 

Sir  Isaac  Newton .94 

Lord  Kelvin  (Sir  William  Thomson) 124 

Hermann  von  Helmholtz       ........  158 

James  Clerk-Maxwell 180 

James  Prescott  Joule 274 

James  Watt 274 

Michael  Faraday 282 

Alessandro  Volta 304 

Georg  Simon  Ohm .  304 

Benjamin  Franklin        .........  314 

Hans  Christian  Oersted 322 

Joseph  Henry         ..........  358 

Sir  William  Crookes 366 

Wilhelm  Konrad  Rontgen 366 

Madame  Curie  '" 370 

Sir  Joseph  John  Thomson 380 

Heinrich  Rudolf  Hertz .390 

Lord  Rayleigh  (John  William  Strutt) 400 


FIRST  PRINCIPLES  OF  PHYSICS 


CHAPTER   I 

INTRODUCTION 

I.     MATTER  AND  ENERGY 

1.  Physics  Defined.  —  Physics  is  often  defined  as  the 
science  of  matter  and  energy.  Matter  is  everything  we  can 
see,  taste,  or  touch  ;  such  as  air,  water,  earth,  gas,  wood, 
steam  —  in  short,  everything  which  occupies  space.  Energy 
is  the  capacity  for  doing  work,  that  is,  for  producing  any 
change  in  the  position  or  condition  of  matter,  especially 
against  resistance  opposing  such  a  change.  Water  in  an 
elevated  tank,  steam  under  pressure  in  a  boiler,  a  flying 
cannon  ball,  —  all  have  the  capacity  for  doing  work,  over- 
coming resistance,  or  changing  the  position  and  motion  of 
bodies. 

So  if  everything  which  we  recognize  by  the  senses  is 
matter,  and  every  change  in  matter  involves  energy,  it  is 
plain  that  physics,  which  is  the  science  of  matter  and  energy, 
is  a  universal  science,  touching  our  life  at  every  point. 
Whatever  we  see  or  touch  is  matter  ;  whatever  we  do 
exhibits  energy.  Countless  physical  phenomena  are  tak- 
ing place  about  us  every  day  :  a  girl  playing  tennis,  a  boy 
rowing  a  boat,  the  school  bell  ringing,  the  sun  giving 
light  and  heat,  the  wind  flapping  a  sail,  an  apple  falling 
from  a  tree,  a  train  or  an  automobile  whizzing  by,  —  all 
are  examples  of  matter  and  associated  energy. 

1 


2  v  v    ^INTRODUCTION 

Physics  is  hofrsa  mubls  iC&ucetii^ct  with  matter  alone  or  with  energy 
alone  as  with  the  relations  of  the  two.  A  baseball  is  of  little  interest 
in  itself;  it  becomes  interesting  only  in  connection  with  a  bat  and 
the  energy  of  the  player's  arm.  The  engine  driver's  interest  is  not  so 
much  in  the  engine  itself  as  in  the  engine  with  steam  up  ready  to 
drive  it.  No  one  would  care  to  purchase  an  automobile  to  stand  in 
a  garage ;  its  attractiveness  lies  in  the  fact  that  it  becomes  almost 
a  living  thing  when  its  motor  is  vitalized  by  the  heat  of  combustion 
of  gasoline. 

2.  Physical  Principles  and  their  Applications.  —  The  phe- 
nomena of  physics  appear  in  everything  we  see  or  do.     It 
will  be  impossible  to  learn  about  all  of  them  in  a  single 
year.     This  is  especially  true  because  the  phenomena  of 
physics,    and   in   particular    the   applications   of   physical 
principles,  are  constantly  changing.     But  the  principles 1 
remain  the  same.     So  this  book  lays  emphasis  on  the  prin- 
ciples of  physics  and  their  common  applications,  leaving  it 
to  the  enthusiasm   and   ingenuity  of   both   teacher  and 
pupils  to  supplement  the  applications  with  illustrations 
drawn  from  life  and  from  scientific  and  technical  journals. 

3.  States  of  Matter.  —  Matter  may  exist  in  three  states, 
exemplified    by    water,    which    may    assume    either    the 
solid,  the  liquid,  or  the  gaseous  form.     Ice,  water,  and 
water  vapor  may  all  exist  together  at  the  same  tempera- 
ture. 

Briefly  described, 
Solids  have  definite  size  and  shape. 

Liquids  have  definite  size,  but  the  shape  is  that  of  the  con- 
taining vessel. 

1  An  hypothesis  is  a  supposition  which  serves  to  explain  phenomena.  The 
more  varied  the  phenomena  it  explains,  the  greater  the  probability  of  its 
truth.  When  the  evidence  in  support  of  it  becomes  large,  it  is  raised  to  the 
rank  of  a  theory ;  and  when  its  truth  is  fully  established,  it  becomes  a  law  or 
principle. 


PROPERTIES   OF  MATTER  3 

G-ases  have  neither  definite  size  nor  shape,  both  depending 
on  the  containing  vessel. 

Some  substances  are  neither  wholly  in  the  one  state  nor 
in  the  other.  Sealing  wax  softens  by  heat  and  passes 
gradually  from  the  solid  to  the  liquid  state.  Shoemaker's 
wax  breaks  into  fragments  like  a  solid  under  the  blow  of 
a  hammer,  but  under  long-continued  pressure  it  flows  like 
a  liquid,  though  slowly,  and  may  be  molded  at  will. 

II.     PROPERTIES   OF  MATTER 

4.  Properties  General  and  Special.  —  The    properties   of 
matter  are  the  qualities  which  serve  to  describe  and  define 
it.     They  are  either  general,  that  is,  common  to  all  kinds 
of  matter;    or  special,  that  is,  found  in  some  kinds  of 
matter  but  not  in  others.     Thus,  all  matter  has  extension, 
or  occupies  space.     On  the  other  hand,  a  piece  of  common 
window  glass  lets  light  pass  through  it,  or  is  transparent, 
while  a  piece  of  sheet  iron  does  not  transmit  light,  or  is 
opaque.     A  watch  spring  recovers  its  shape  after  bending, 
or  is  elastic,  while  a  strip  of  lead  possesses  this  property 
in  so  slight  a  degree  that  it  is  classed  as  inelastic.     So  we 
see  that  while  extension  is  a  general  property  of  matter, 
transparency  and  elasticity  are  special  properties. 

A.    General  Properties 

5.  Extension.  —  All  bodies  have  three  dimensions,  length, 
breadth,  and  thickness.     A  sheet  of  tissue  paper  or  of  gold 
leaf,  at  first  thought,  appears  to  have  but  two  dimensions, 
length  and  breadth;    but  while  its  third   dimension   is 
relatively  small,  if  its  thickness  should  actually  become 
zero,  it  would  cease  to  be  either  a  sheet  of  paper  or  a  piece 
of  gold  leaf.     Extension  is  the  .property  of  occupying  space 
or  having  dimensions. 


INTRODUCTION 


6.  Impenetrability.  —  Matter  occupies  space,  but  no  two 
portions  of  matter  can  occupy  the  same  space  at  the  same 
time.  The  volume  of  an  irregular 
solid,  such  as  a  lump  of  coal,  may 
be  measured  by  noting  the  volume 
of  liquid  displaced  when  the  solid 
is  completely  immersed  in  it.  The 
general  property  of  matter  that  no 
two  bodies  can  occupy  the  same  space 
at  the  same  time  is  known  as  impene- 
trability. 

Put  a  lump  of  coal  into  a  tall  graduate 
partly  filled  with  water,  as  in  Fig.  1.  Note 
the  reading  at  the  surface  of  the  water 
before  and  after  putting  in  the  coal;  the 

difference  is  the  volume  of  the  water  displaced,  or  that  of  the  piece 

of  coal. 


7.  Inertia.  —  The 
most  characteristic  gen- 
eral property  of  matter 
is  inertia.  Inertia  is 
the  property  which  all 
matter  possesses  of  re- 
sisting any  attempt  to 
start  it  if  at  rest,  to  stop 
it  if  in  motion,  or  to 
change  either  the  direc- 
tion or  the  amount  of  its 
motion.  If  a  moving 
body  stops,  its  arrest  is 
always  owing  to  some- 
thing outside  of  itself  ; 
and  if  a  body  at  rest  is 


Fig.  2 


PROPERTIES  OF  MATTER 


set  moving,  motion  must  be 
imparted  to  it  by  some  other- 
body. 

8.    Illustrations  of  Inertia.  - 

Many   familiar    facts    are    due   to 

inertia.     When  a  street  car  stops 

suddenly,  a  person  standing  con- 
tinues by  inertia  to  move  forward, 

or  is  apparently  thrown  toward  the 

front  of  the  car ;  the*  driver  of  a 

racing    motor    car    is    apparently 

thrown    with    violence   when    the 

rapidly  moving  car  collides  with  a 

post  or    a  tree ;    the    fact  is    the 

car  is  violently  stopped,  while  the 

driver  continues  to  move  forward 

as  before  the  collision.     When  a 

fireman  shovels  coal  into  a  furnace, 

he  suddenly  arrests  the  motion  of 

the  shovel  and  leaves  the  coal  to 

move  forward  by  inertia.    A  smooth 

cloth  may  be  snatched  from  under 

a  heavy  dish  without  disturbing  it.     The  violent  jar  to  a  water  pipe 
when  a  faucet  is  suddenly  closed  is  accounted  for  by 
the  inertia  of  the  stream.     The  persistence  with  which 
a  spinning  top  maintains  its  axis  of  rotation  in  the' 
same  direction  is  due  to  its  inertia. 

Bren nan's  monorail  car  (Fig.  2)  is  kept  in  an  up- 
right position  by  the  inertia  of  a  rapidly  revolving 
wheel  of  great  weight.  Tall  columns,  chimneys,  and 
buildings  are  sometimes  twisted  around  by  violent 
earthquake  movements  (Fig.  3).  The  sudden  circu- 
lar motion  of  the  earth  under  a  column  leaves  it 
standing  still,  while  the  slower  return  motion  carries 
it  around. 

Suspend  a  heavy  weight  by  a  cotton  string,  as  in 

Fig.  4,  and  tie  a  piece  of  the  same  string  to  the  under  side  of  the 

weight.     A  steady  downward  pull  at  B  will  break  the  upper  string 


Fig.  3 


Fig.  4 


6  INTRODUCTION 

because  it  carries  the  greater  load.  A  sudden  downward  pull  on  B 
will  break  the  lower  string  before  the  pull  reaches  the  upper  one  on 
account  of  the  inertia  of  the  weight. 

9.  Mass.  —  Another  general  property  of  matter  is  mass. 
We  are  all  familiar  with  the  fact  that  the  less  matter 
there  is  in  a  Nbody,  the  more  easily  it  is  moved,  and  the 
more  easily  it  is  stopped  when  in  motion.     One  can  tell 
an  empty  barrel  from  a  full  one  by  a  kick,  a  block  of  wood 
from  a  brick  by  shoving  it  with  the  foot,  and  a  tennis  ball 
from  a  baseball  by  catching  it.     Mass  is  the  measure  of 
the  resistance  which  a  body  offers  to  motion  or  change  of 
motion;  it  is  therefore  the  measure  of  the  body's  inertia. 
Mass  must  not  be  confused  with  weight  (§  116)  because 
mass  is  independent  of  gravity.     The  mass  of  a  meteoric 
body  is  the  same  when  flying  through  space  as  when  it 
strikes  the  earth  and  embeds  itself  in  the  ground.     If  it 
could  reach  the  center  of   the   earth,  its  weight  would 
become  zero ;  at  the  surface  of  the  sun  it  would  weigh 
nearly  twenty-eight  times  as  much  as  at  the  earth's  sur- 
face ;  but  its  mass  would  be  the  same  everywhere.     For 
this  reason,  and  others  which  will  appear  later,  in  discuss- 
ing the  laws  of  physics  we  prefer  to  speak  of  mass  when  a 
student  thinks  the  term  weight  might  be  used  as  well. 

10.  Cohesion  and  Adhesion.  —  All  bodies  are  made  up  of 
very  minute  particles,  which  are  separately  invisible,  and 
are  called  molecules.      Cohesion  is  the  force  of  attraction  be- 
tween molecules,  and  it  binds  together  the  molecules  of  a 
substance  so  as  to  form  a  larger  mass  than  a  molecule. 
Adhesion  is  the  force  uniting  bodies  by  their  adjacent  sur- 
faces.    When   two  clean  surfaces  of  white-hot  wrought 
iron  are  brought  into  close  contact  by  hammering,  they 
cohere  and  become  a  single  body.      If  a  clean  glass  rod  be 
dipped  into  water  arid  then  withdrawn,  a  drop  will  adhere 


PROPERTIES   OF  MATTER 


•J 


to'  it*  Glue,  adhesive  plaster,  and  postage  stamps  stick 
by  adhesion.  Mortar  adheres  to  bricks  and  gold  plating 
to  brass. 

Suspend  from  one  of  the  scalepans  of  a  beam  balance  a  clean  glass 
disk  by  means  of  threads  cemented  to  it  (Fig.  5).  After  counterpois- 
ing the  disk,  place  below  it  a  vessel  of  water,  and  adjust  so  that  the 
disk  just  touches  the  surface  of  the  water  when  the  beam  of  the  bal- 
ance is  horizontal.  Now  add  weights  to  the  opposite  pan  until  the 
disk  is  pulled  away  from  the  water.  Note  that  the  under  surface  of 
the  disk  is  wet.  The  adhesion  of  the  water  to  the  glass  is  greater 
than  the  cohesion  between 
the  molecules  of  the  water. 
If  mercury  be  substituted  for 
water,  a  much  greater  force 
will  be  necessary  to  separate 
the  disk  from  the  mercury, 
but  no  mercury  will  adhere 
to  it.  The  force  of  cohe- 
sion between  the  molecules 
of  the  mercury  is  greater 
than  the  adhesion  between 
it  and  the  glass. 

Cut  a  fresh,  smooth  sur- 
face on  two  lead  bullets  and 
hold  these  surfaces  gently  to- 
gether. They  will  not  stick. 

Now  press  them  tightly  together  with  a  slight  twisting  motion.  They 
will  adhere  quite  firmly.  This  fact  shows  that  molecular  forces  act 
only  through  insensible  distances.  It  has  been  shown  that  they  vanish 
in  water  at  a  range  of  about  one  five-hundred-thousandth  of  an  inch. 

An  interesting  example  of  selective  adhesion  occurs  in  the  winning 
of  diamonds  in  South  Africa.  The  mixed  pebbles  and  other  worth- 
less stones,  with  an  occasional  diamond,  are  washed  down  an  inclined 
shaking  surface  covered  with  grease.  Only  the  diamonds  and  a  few 
other  precious  stones  stick  to  the  grease ;  the  rest  are  washed  away. 

11.  Porosity.  —  Sandstone,  unglazed  pottery,  and  similar 
bodies  absorb  water  without  change  in  volume.  The  water 


Fig.  5 


8  INTRODUCTION 

fills  the  small  spaces  called  pores,  which  are  visible  either 
to  the  naked  eye  or  under  a  microscope.  All  matter  is 
probably  porous,  though  the  pores  are  invisible,  and  the 
corresponding  property  is  called  porosity.  In  a  famous 
experiment  in  Florence  many  years  ago,  a  hollow  sphere 
of  heavily  gilded  silver  was  filled  with  water  and  put 
under  pressure.  The  water  exuded  through  the  pores  of 
the  silver  and  gold  and  stood  in  beads  on  the  surface. 
Francis  Bacon  observed  a  similar  phenomenon  with  a  lead 
sphere. 

Oil  penetrates  into  marble  and  spreads  through  it.  Even  so  dense 
a  substance  as  agate  is  porous,  for  it  is  artificially  colored  by  the  ab- 
sorption, first  of  one  liquid  and  then  of  another  which  acts  chemically 
on  the  first ;  the  result  is  a  deposit  of  coloring  matter  in  the  pores  of 
the  agate. 

B.   Special  Properties 

12.  Tenacity  and  Tensile  Strength.  —  Tenacity  is  the  resist- 
ance which  a  body  offers  to  being  torn  apart.  The  tensile 
strength  of  wires  is  tested  by  hanging  them  vertically  and 
loading  with  successive  weights  until  they  break  (Fig.  6). 
3  The  breaking  weights  for  wires  of  different  mate- 
rials but  of  the  same  cross  section  differ  greatly. 
A  knowledge  of  tensile  strength  is  essential  in  the 
design  of  telegraph  wires  and  cables,  suspension 
bridges,  and  the  tension  members  of  all  steel 
structures. 

Tenacity  diminishes  with  the  duration  of  the 
pull,  so  that  wires  sometimes  break  with  a  load 
which  they  have  supported  for  a  long  time.  Lead 
has  the  least  tenacity  of  all  solid  metals,  and  cast 
steel  the  greatest.  Even  the  latter  is  exceeded  by 
fibers  of  silk  and  cotton.  Single  fibers  of  cotton 
Fig.  6  can  support  millions  of  times  their  own  weight. 


PROPERTIES    OF   MATTER 

13.  Ductility. — Ductility  is  the  property  of  a  substance 
which  permits  it  to  be  drawn  into  wires  or  filaments.     Gold, 
copper,  silver,  and  platinum  are  highly  ductile.     The  last 
is  the  most  ductile  of  all.     It  has  been  drawn  into  wire 
only  0.00003  inch  in  diameter.     A  mile  of  this  wire  would 
weigh  only  1.25  grains. 

Other  substances  are  highly  ductile  only  at  high  tem- 
peratures. Glass  has  been  spun  into  such  fine  threads 
that  a  mile  of  it  would  weigh  only  one  third  of  a  grain. 
Melted  quartz  has  been  drawn  into  threads  not  more  than 
0.00001  inch  in  diameter.  Such  threads  have  nearly  as 
great  tenacity  as  steel'. 

14.  Malleability.  —  Malleability  is  a  property  which  per- 
mits of  hammering  or  rolling  some  metals  into  thin  sheets. 
Gold  leaf,  made  by  hammering  between  skins,  is  so  thin 
that  it  is  partially  transparent  and  transmits  green  light. 
Zinc  is  malleable  when  heated  to  a  temperature  of  from 
100°  to  150°  C.  (centigrade  scale).     It  can  then  be  rolled 
into   sheets.      Nickel   at    red   heat   can   be    worked   like 
wrought  iron.     Malleable  iron  is  made  from  cast  iron  by 
heating  it  for  several  days  in  contact  with  a  substance 
which  removes  some  of  the  carbon  from  the  cast  iron. 

15.  Hardness  and  Brittleness.  — Hardness  is  the  resistance 
offered  by  a  body  to  scratching  by  other  bodies.     The  relative 
hardness  of  two  bodies  is  ascertained  by  finding  which  will 
scratch  the  other.     Diamond  is  the  hardest  of  all  bodies 
because  it  scratches  all  others.     Sir  William  Crookes  has 
shown  that  diamonds  under  great  hydraulic  pressure  be- 
tween mild  steel  plates  completely  embed  themselves  in 
the  metal.     Carborundum,  an  artificial  material  used  for 
grinding  metals,  is  nearly  as  hard  as  diamond. 

Brittleness  is  aptness  to  break  under  a  blow.     It  must  be 


10  INTRODUCTION 

distinguished  from  hardness.  Steel  is  hard  and  tough, 
while  glass  is  hard  and  brittle. 

Tool  steel  becomes  glass  hard  and  brittle  when  suddenly 
cooled  from  a  high  temperature.  The  tempering  of  steel 
is  the  process  of  giving  the  degree  of  hardness  required 
for  various  purposes.  It  consists  usually  in  first  plunging 
the  article  at  red  heat  into  cold  water  or  other  liquid  to 
give  it  an  excess  of  hardness  ;  then  reheating  gradually 
until  the  hardness  is  reduced,  or  "drawn  down,"  to  the 
required  degree.  The  indication  of  the  hardness  is  the 
color  appearing  on  a  polished  portion,  such  as  straw- 
yellow,  brown-yellow,  purple,  or  blue. 

The  process  of  annealing  as  applied  to  iron  and  glass  is 
used  to  render  them  less  brittle.  It  is  done  by  cooling 
very  slowly  and  uniformly  from  a  high  temperature.  Soft 
iron  is  thus  made  more  ductile,  while  glass  is  relieved 
from  the  molecular  stresses  set  up  in  rapid  cooling,  and 
it  thus  becomes ,. tougher  and  more  uniform.  The  best 
lamp  chimneys  are  annealed  by  the  manufacturer.  Disks 
of  glass  for  telescope  lenses  must  be  carefully  annealed  to 
prevent  fracture  and  warping  during  the  process  of  grind- 
ing and  polishing. 

III.     PHYSICAL  MEASUREMENTS 

16.  Units.  — To  measure  any  physical  quantity  a  certain 
fixed  amount  of  the  same  kind  of  quantity  is  used  as  the 
unit.  For  example,  to  measure  the  length  of  a  body,  some 
arbitrary  length,  such  as  a  foot,  is  chosen  as  the  unit  of 
length;  the  length  of  a  body  is  the  number  of  times  this  unit  is 
contained  in  the  longest  dimension  of  the  body.  The  unit  is 
always  expressed  in  giving  the  magnitude  of  any  physical 
quantity;  the  other  part  of  the  expression  is  the  numerical 
value.  For  example,  60  (feet),  500  (pounds), 45  (seconds). 


PHYSICAL  MEASUREMENTS  11 

In  like  manner,  to  measure  a  surface  the  unit,  or  stand- 
ard surface,  must  be  given,  such  as  a  square  foot ;  and  to 
measure  a  volume,  the  unit  must  be  a  given  volume,  such, 
for  example,  as  a  cubic  inch,  a  quart,  or  a  gallon. 

17.  Units  of  Length.  —  The  two  systems  of  units  in 
common  use  are  the  metric  and  the  English.  In  the  for- 
mer the  unit  of  length  is  the  meter.  It  is  the  distance 


Fig.  7 

between  two  transverse  lines  on  each  of  two  bars  of  plati- 
num-iridium  at  the  temperature  of  melting  ice.  These 
bars,  called  national  prototypes,  are  preserved  at  the 
Bureau  of  Standards  in  Washington  (Fig.  7). 

The  meter  (m.)  is  divided  into  10  decimeters  (dm.),  the 
decimeter  into  10  centimeters  (cm.),  and  the  centimeter 
into  10  millimeters  (mm.).  The  only  multiple  of  the 
meter  in  general  use  is  the  kilometer  (km.),  equal  -to  1000 
meters.  It  is  used  to  measure  such  distances  as  are 
expressed  in  miles  in  the  English  system. 

In  the  metric  system  areas  are  measured  in  square 
millimeters  (mm.2),  square  centimeters  (cm.2),  etc.  In 
like  manner  volumes  are  measured  in  cubic  millimeters 
(mm.3),  cubic  centimeters  (cm.3),  etc.  The  cubic  deci- 
meter (dm.3)  is  called  a  liter  (1.)  ;  it  is  equal  to  1000  cm.3. 

The  weights  and  measures  in  common  use  in  the  United 
States  were  denned  by  Act  of  Congress  in  1866  in  terms 
of  those  of  the  metric  system.  By  this  act  the  legal  value 
of  the  yard  is  f  f  $$  of  a  meter ;  conversely,  the  meter  is 
39.37  inches.  The  inch  is,  therefore,  2.540  cm.  The 


INTRODUCTION 


relation  between  the  centimeter  scale  and  the  inch  scale 
is  shown  in  Fig.  8. 

100  MILLIMETERS  =  10  CENTIMETERS  =  1   DECIMETER  =  3. 937  INCHES. 


1 

2 

3                        4]                      5 

6]                       7                        8                        9|                    10| 

I  I  I  I  I  I  I  I  II  I  II    I  I  II  I  I  !  !  I    I  I  I  I  !  I  I  I    I  I  I  I  I  I  I  I  I    I 

I 

i 

i      I 

I 

II             I             I      I       I      I 

I 

I     I     |   -I     I     I     I          till          111 

t 

2 

3                                                              4 

INCHES  AND  TENTHS 

Fig.  8 

The  unit  of  length  in  the  English  system  for  the  United 
States  is  the  yard,  defined  as  above.  One  third  of  the 
yard  is  the  foot,  and  one  thirty-sixth  is  the  inch.  The 
gallon  of  231  cubic  inches  is  the  unit  of  volume  for  liquid 
measure.  Tables  for  the  conversion  of  quantities  from 
one  system  of  units  into  the  other  may  be  found  in 
Appendix  II. 

18.  Units  of  Mass.  —  The  unit  of  mass  in  the  metric 
system  is  the  kilogram  (kgm.).  The  United  States  has 

two  prototype  kilograms 
made  of  platinum-irid- 
ium  and  preserved  at 
the  Bureau  of  Stand- 
ards in  Washington 
(Fig.  9).  The  gram 
(gm.)  is  one  thousandth 
of  the  kilogram.  The 
latter  was  originally  de- 
signed to  represent  the 
mass  of  a  liter  of  pure 
water  at  4°  C.  (centi- 
grade scale).  For  prac- 
tical purposes  this  is  the 
Fig.  9  kilogram.  The  gramas 


PHYSICAL  MEASUREMENTS  13 

therefore  equal  to  the  mass  of  a  cubic  centimeter  of  water 
at  the  same  temperature.  The  mass  of  a  given  body  of 
water  can  thus  be  immediately  inferred  from  its  volume. 

The  unit  of  mass  in  the  English  system  is  the  avoirdu? 
pois  pound  (lb.)..  The  ton  of  2000  pounds  is  its  chief 
multiple;  its  submultiples  are  the  ounce  (oz.)  and  the 
grain  (gr.).  The  avoirdupois  pound  is  equal  to  16  ounces 
and  to  7000  grains.  The  "  troy  pound  of  the  mint "  con- 
tains 5760  grains.  In  1866  the  mass  of  the  5-cent  nickel 
piece  was  legally  fixed  at  5  grams;  and  in  1873  that  of 
the  silver  half  dollar  at  12.5  grams.  One  gram  is  equal 
approximately  to  15.432  grains.  A  kilogram  is  very 
nearly  2.2  pounds. 

19.  The  Unit  of  Time.  —  The  unit  of  time  in  universal 
use  in  physics  is  the  second.     It  is  -g^TTo  of  a  mean  solar 
day.     The  number  of  seconds  between  the  instant  when 
the  sun's  center  crosses  the  meridian  of  any  place  and  the 
instant  of  its  next  passage  over  the  same  meridian  is  not 
uniform,  chiefly  because  the  motion  of  the  earth  in  its  orbit 
about  the  sun  varies  from  day  to  day.     The  mean  solar 
day  is  the  average  length  of  all  the  variable  solar  days 
throughout  the  year.     It  is  divided  into  24  x  60  x  60  = 
86,400  seconds  of  mean  solar  time,  the  time  recorded  by 
clocks  and  watches.     The  sidereal  day  used  in  astronomy 
is  nearly  four  minutes  shorter  than  the  mean  solar  day. 

20.  The  Three  Fundamental  Units.  —  Just  as.  the  meas- 
urement of  areas  and  of  volumes  reduces  simply  to  the 
measurement  of  lengths,  so  it  has  been  found  that  the 
measurement  of  most  other  physical  quantities,  such  as 
the  speed  of  a  ship,  the  pressure  of  water  in  the  mains, 
the  energy  consumed  by  an  electric  lamp,  and  the  horse 
power  of  an  engine,  may  be  made  in  terms  of  the  units  of 


14  INTRODUCTION 

length,  mass,  and,  time.  For  this  reason  these  three  are 
considervtTfundamental  units  to  distinguish  them  from  all 
others,  which  are  called  derived  units. 

The  system  now  in  general  use  in  the  physical  sciences 
employs  the  centimeter  as  the  unit  of  length,  the  gram  as 
the  unit  of  mass,  and  the  second  as  the  unit  of  time.  It 
is  accordingly  known  as  the  c.  g.  s.  (centimeter-gram- 
second)  system. 

Exercises  and  Problems 

Problems  involving  the  relations  between  the  two  systems  of 
measurement  should  be  solved,  using  the  exact  values  of  §§  17  and  18. 

1.  Could  the  volume  of  a  lump  of  sugar  be  determined  by  the 
method  of  §6?  f 

2.  The  circus  rider  standing  on  the  back  of  a  moving  horse  jumps 
straight  up  and  not  forward  in  order  to  go  through  a  paper-covered 
hoop.     Explain. 

3.  An  ax  handle  is  driven  into  the  ax  by  pounding  the  end  of 
the  handle  rather  than  the  ax.     Explain. 

4.  To  keep  from  falling  the  conductor  runs  by  the  side  of  the 
moving  car  before  he  jumps  on.     Why? 

5.  A  man    standing   on  a  moving  car  jumps  vertically  upward. 
Does  he  come  down  on  the  spot  from  which  he  jumped,  or  back  of  it  ? 

6.  Why  will  a  bullet  from  a  rifle  shot  at  a  window  cut  a  smooth 
hole  through  the  glass,  but  if  thrown  against  it  by  hand  shatter  it? 

7.  Lead  bullets  fired  from  a  rifle  against  a  thick  plate  of  lead  are 
found  welded  to  the  plate.     Why  ? 

8.  If  a  horse  in  starting  a  loaded  wagon  is  permitted  to  jump,  he 
is  likely  to  break  the  harness ;  but  if  he  pulls  steadily,  the  harness  is 
sufficient  to  stand  the  pull.     Explain. 

9.  Why  does  the  flywheel  cause  an  engine  to  run  more  uniformly  ? 

10.  Why  does  a  pendulum  keep  moving  when  the  bob  reaches  the 
lowest  point  of  its  swing? 

11.  How  many  meters  in  a  rod  ? 


EXEECISES  AND  PROBLEMS  15 

12.  The  Washington  monument  is  555  ft.  high.     Find  its  height 
in  meters. 

13.  How  many  gallons  in  100  liters? 

14.  Calculate  the  number  of  liters  in  10  gal. 

15.  •  A  motor  boat  has  a  speed  of  20  mi.  an  hour.     Express  it  in 
kilometers  per  hour. 

16.  At   10?  per  pound,  what  will  be  the  cost   of   10   kgrn.  of 
rice? 

17.  A  cubic  foot  of  water  weighs  62.4  Ib. ;  what  is  the  weight  of  a 
cubic  inch  in  grains? 

18.  A  cubic  foot  of  water  weighs  62.4  Ib. ;  what  is  the  weight  of  a 
cubic  inch  in  grams? 

19.  Find  the  difference  in  centimeters  between  one  foot  and  30  cm. 

20.  The  greatest  allowable  weight  of  a  package  by  foreign  parcels 
post  is  5  kgm.     What  is  the  nearest  whole  number  of  pounds? 


CHAPTER   II 

MOLECULAR  PHYSICS 

I.     MOLECULAR  MOTION 

21.  Diffusion  of  Gases.  —  If  two  gases  are  placed  in  free 
communication  with  each  other  and  are  left  undisturbed, 
they  will  mix  rather  rapidly.  Even  though  they  differ  in 
density  and  the  heavier  gas  is  at  the  bottom,  the  mixing 
goes  on.  This  process  of  the  spontaneous  mixing  of  gases 
is  called  diffusion. 

The  rapidity  with  which  gases  diffuse  may  be  illus- 
trated by  allowing  illuminating  gas  to  escape  into  a  room, 
or  by  exposing  ammonia  in  an  open  dish.  The  odor  quickly 
reveals  the  presence  of  either  gas  in 
all  parts  of  the  room,  even  when  air 
currents  are  suppressed  as  far  as  pos- 
sible. A  more  agreeable  illustration 
is  furnished  by  a  bottle  of  smelling 
salts.  If  it  is  left  open,  the  perfume 
soon  pervades  the  whole  room. 

Fill  one  of  a  pair  of  jars  (Fig.  10)  with 
P.        _  the  fumes  of  strong  hydrochloric  acid,  and 

the  other  with  gaseous  ammonia,  and  place 

over  them  the  glass  covers.  Bring  the  jars  together  as  shown,  and 
after  a  few  seconds  slip  out  the  cover  glasses.  In  a  few  minutes  both 
jars  will  be  filled  with  a  white  cloud  of  the  chloride  of  ammonia. 
Instead  of  these  vapors  air  and  illuminating  gas  may  be  used,  and 
after  diffusion,  the  presence  of  an  explosive  mixture  in  both  jars  may 
be  shown  by  applying  a  flame  to  the  mouth  of  each  separately. 

16 


MOLECULAR  MOTION  17 

22.  Effusion  through  Porous  Walls.  —  The  passage  of  a 
gas  through  the  pores  of  a  solid  is  known    'ds  Affusion. 
The  rate  of  effusion  for  different  gases  is  nearly  inversely 
proportional   to   the   square   of   their   relative    densities. 
Hydrogen,  for  example,  which  is  one  six- 
teenth as  heavy  as  oxygen,  passes  through 

very   small    openings   four   times   as   fast 
as  oxygen. 

Cement  a  small  unglazed  battery  cup  to  a  funnel 
tube,  and  connect  the  latter  to  a  flask  nearly  filled 
with  water  and  fitted  with  a  jet  tube,  as  shown  in 
Fig.  11.  Invert  over  the  porous  cup  a  large  glass 
beaker  or  bell  jar,  and  pass  into  it  a  stream  of 
hydrogen  or  illuminating  gas.  If  all  the  joints  are 
air-tight,  a  small  water  jet  will  issue  from  the  fine 
tube.  The  hydrogen  passes  freely  through  the 
invisible  pores  in  the  walls  of  the  porous  cup  and 
produces  gas  pressure  in  the  flask.  If  the  beaker  pj~  j  j 

is  now  removed,  the  jet  subsides  and  the  pressure 
in  the  flask  quickly  falls  to  that  of  the  air  outside  by  the  passage  of 
hydrogen  outward  through  the  pores  of  the  cup. 

23.  Molecular  Motion  in  Gases.  — The  simple  facts  of  the 
diffusion  and  effusion  of  gases  lead  to  the  conclusion  that 
their  molecules  are  not  at  rest,  but  are  in  coiisl^tjim 
rapid  motion.     The  property  of  indefinite  expansibility  is 
a  further  evidence   of   molecular   motion  in  gases.     No 
matter  how  far  the  exhaustion  is  carried  by  an  air  pump, 
the  gas  remaining  in  a  closed  vessel  expands  and  fills  it. 
This  is  not  due  to  repulsion  between  the  molecules,  but 
to  their  motions.     Gases  move  into  a  good  vacuum  much 
more  quickly  than  they  diffuse  through  one  another.     In 
diffusion  their  motion  is  frequently  arrested  by  molecular 
collisions,  and  hence  diffusion  is  impeded. 

The  property  of  rapid  expansion  into  a  free  space  is 


18  MOLECULAR  PHYSICS 

a  highly  important  one.  Witness  the  operation  of  a  gaso- 
line engine,  in  which  the  inlet  valve  presents  only  a  nar- 
row opening  for  a  small  fraction  of  a  second  ;  and  yet  this 
brief  period  suffices  for  the  explosive  mixture  to  enter  and 
fill  the  cylinder. 

24.  Pressure  Produced   by  Molecular  Bombardment.  —  It 

would  be  possible  to  keep  an  iron  plate  suspended  hori- 
zontally in  the  air  by  the  impact  of  a  great  many  bullets 
fired  up  against  its  under  surface.  The  clatter  of  an  in- 
definitely large  number  of  hailstones  on  a  roof  forms  a 
continuous  sound,  and  their  fall  beats  down  a  field  of 
grain  flat  to  the  ground.  So  the  rapidly  moving  molecules 
of  a  gas  strike  innumerable  minute  blows  against  the  walls 
of  the  containing  vessel  and  these  blows  compose  a  contin- 
uous pressure.  This,  in  brief,  is  the  kinetic  theory  of  the 
pressure  of  a  gas. 

25.  The  Velocity  of  Molecules.  — It  has  been  found  pos- 
sible to  calculate  the  velocity  which  the  molecules  of  air 
must  have  under  standard  conditions  to  produce  by  their 
impact  against  the  walls  of  a  vessel  the  pressure  of  one 
atmosphere,  or  1033  gm.  per  square  centimeter.    It  is  about 
450  m.  per  second.     For  the  same  pressure  of  hydrogen, 
which  is  only  one  fourteenth  as  heavy  as  air,  the  velocity 
has  the  enormous  value  of  1850  m.  per  second.     The  high 
speed  of  the  hydrogen  molecules  accounts  for  their  rela- 
tively rapid  progress  through  porous  walls. 

26.  Diffusion  of  Liquids.  —  Liquids  diffuse  into  one  an- 
other in    a    manner   similar   to   that   of    gases,   but   the 
process   is  indefinitely   slower.     Diffusion  in  liquids,  as 
in   gases,   shows    that   the    molecules    have   independent 
motion  because  they  move  more  or  less  freely  among  one 
another. 


8  UEFA  CE  PHENOMENA  19 

Let  a  tall  jar  be  nearly  filled  with  water  colored  with  blue 
litmus,  and  let  a  little  strong  sulphuric  acid  be  introduced  into 
the  jar  at  the  bottom  by  means  of  a  thistle  tube  (Fig.  12). 
The  density  of  the  acid  is  1.8  times  that  of  the  litmus  solu- 
tion, and  the  acid  therefore  remains  at  the  bottom  with  a 
well-defined  surface  of  separation,  which  turns  red  on  the 
litmus  side  because  acid  reddens  litmus.  But  if  the  jar 
be  left  undisturbed  for  a  few  hours,  the  line  of  separation 
will  lose  its  sharpness  and  the  red  color  will  move  gradually 
upward,  showing  that  the  acid  molecules  have  made  their 
way  toward  the  top.  Fig-  1 

27.  Diffusion  of  Solids.  —  The  diffusion  of  solids  is  much 
less  pronounced  than  the  diffusion  of  gases  and  liquids,  but 
it  is  known  to  occur.     Thus,  if  gold  be  overlaid  with  lead, 
the  presence  of  gold  throughout  the  lead  may  in  time  be 
detected.     Mercury  appears  to  diffuse   through   lead   at 
ordinary  temperatures  ;    in  electroplating  the  deposited 
metal  diffuses  slightly  into  the  baser  metal ;    at  higher 
temperatures  metals  diffuse  into  one  another  to  a  marked 
degree,  so  that  there  is  evidence  of  molecular  motion  in 
solids  also. 

II.    SURFACE  PHENOMENA 

28.  Molecular  Forces  in  Liquids.  —  A  primitive  idea  of 
force  is  derived  from  the  sense  of  muscular  exertion  in 
lifting  a  weight,  pushing  a  cart,  stretching  a  spring,  catch- 
ing a  ball,  throwing  a  stone,  or  making  intense  muscular 
effort  in  running  at  top  speed.     By  an  easy  transition  of 
ideas  we  carry  this  conception  over  to  forces  other  than 
those  exerted  by  men  and  animals,  such  as  those  between 
the  molecules  of  a  body.     Molecular  forces  act  only  through 
insensible  distances,  such  as  the  distances  separating  the 
molecules  of  solids  and  liquids.     A  clean  glass  rod  does 
not  attract  water  until  there  is  actual  contact  between  the 


20 


MOLECULAR  PHYSICS 


two.     If  the  rod  touches  the  waiter,  the  latter  clings  to 

the  glass,  and  when  the  rod  is  withdrawn,  a  drop  adheres 
to  it.  If  the  drop  is  large  enough, 
its  weight  tears  it  away,  and  it  falls 
as  a  little  sphere. 

By  means  of  a  pipette  a  large  glob- 
ule of  olive  oil  may  be  introduced 
below  the  surface  of  a  mixture  of 
water  and  alcohol,  the  mixture  having 

been  adjusted  to  the  same  density  as  that  of  the  oil  by 

varying  the  proportions.    The  globule  then  assumes  a  truly 

spherical  form  and  floats  anywhere  in  the  mixture  (Fig.  13). 
Cover  a  smooth   board  with  fine  dust,  such  as  lyco- 

podium  powder  or  powdered  charcoal.     If  a  little  water 

be    dropped    upon 

it   from    a    height 

of  about  two  feet, 

it    will    scatter 

and  take  the  form       ^"^ 

of    little     spheres 

(Fig.  14). 

In  all  these  illustrations  the  spherical  form  is  accounted 

for  by .  the  forces  between  the  molecules  of  the  liquid. 

They  produce  uniform  molecular  pressure  and  form  little 

spheres,  because  a  spherical  surface  is  the  smallest  that 

will  inclose  the  given  volume. 

29.    Condition  at  the  Surface  of  a  Liquid.  —  Bubbles  of  gas  re- 
leased in  the  interior  of  a  cold  liquid  and  rising  to  the  surface  often 
show   some  difficulty  in  breaking  through.     A 
sewing  needle  carefully  placed  on  the  surface  of 
water  floats.      The  water  around  the  needle  is 
EEEE      depressed  and  the  needle  rests  in  a  little  hollow 

(Fig.  15). 
Fig.  15  Let   two  bits  of  wood  float  on  water  a  few 


SUE  FACE  PHENOMENA  21 

millimeters  apart.  If  a  drop  of  alcohol  is  let  fall  on  the  water  be- 
tween them,  they  suddenly  fly  apart. 

A  thin  film  of  water  may  be  spread  evenly  over  a  chemically  clean 
glass  plate ;  but  if  the  film  is  touched  with  a  drop  of  alcohol  on  a  thin 
glass  rod,  the  film  will  break,  the  water  retiring  and  leaving  a  dry 
area  around  the  alcohol. 

The  sewing  needle  indents  the  surface  of  the  water  as  if 
the  surface  were  a  tense  membrane  or  skin,  and  tough  enough 
to  support  the  needle.  This  surface  skin  is  weaker  in  alcohol 
than  in  water  ;  hence  the  bits  of  wood  are  pulled  apart  and 
the  water  is  withdrawn  from  the  spot  weakened  with  alcohol. 

30.  Surface   Tension.  —  The   molecules   composing   the 
surface  of  a  liquid  are   not  under  the   same  ^conditions 
of  equilibrium  as  those  within  the  B 
liquid.      The    latter    are   attracted 

equally  in  all  directions  by  the  sur- 
rounding molecules, -while  those  at 
the  surface  are  attracted  downward 
and  laterally,  but  not  upward  (Fig. 
16).  The  result  is  an  unbalanced 
molecular  force  toward  the  interior  lg*  lc 

of  the  liquid,  so  that  the  surface  layer  is  compressed 
and  tends  to  contract.  The  contraction  means  that  the 
surface  acts  like  a  stretched  membrane,  which  molds  the 
liquid  into  as  small  a  volume  as  possible.  Liquids  in 
small  masses,  therefore,  always  tend  to  become  spherical. 

31.  Illustrations.  —  Tears,   dewelrpps,  and   drops  of   rain  are 
spherical  because  of  the  tension  in  the  surface  film.     Surface  tension 
rounds  the  end  of  a  glass  rod  or  stick  of  sealing  wax  when  softened 
in  a  flame.     It  breaks  up  a  small  stream  of  molten  lead  into  little 
sections,  and  molds  them  into  spheres  which  cool  as  they  fall  and 
form  shot.     Small  globules  of  mercury  on  a  clean  glass  plate  are 
slightly  flattened  by  their  weight,  but  the  smaller  the  globules  the 
more  nearly  spherical  they  are. 


22 


MOLECULAR  PHYSICS 


Fig.  17 


Make  a  stout  ring  three  or  four  inches  in  diameter  with  a  handle 
(Fig.  17).     Tie  to  it  a  loop  of  soft  thread  so  that  the  loop  may  hang 

near  the  middle  of  the  ring. 
Dip  the  ring  into  a  soap  solution 
containing  glycerine,  and  get  a 
plane  film.  The  thread  will  float 
in  it.  Break  the  film  inside  the 
loop  with  a  warm  pointed  wire, 
and  the  loop  will  spring  out 
into  a  circle.  The  tension  of 
the  film  attached  to  the  thread 
pulls  it  out 
equally  in  all 
directions. 

Interesting  surfaces  may  be  obtained  by  dipping 
skeleton  frames  made  of  stout  wire  into  a  soap 
solution.  The  films  in  Fig.  18  are  all  plane,  and 
the  angles  where  three  surfaces  meet  along  a  line 
are  necessarily  120°  for  equilibrium. 

A  bit  of  gum  camphor  on  warm  water,  quitr> 
free  from  an  oily  film,  will  spin  around  in  a 
most  erratic  manner.  The  camphor  dissolves  unequally  at  different 

points,  and  thus  pro- 
duces unequal  weaken- 
ing of  the  surface 
tension  in  different  di- 
rections. 

Make  a  tiny  wooden 
boat  and  cut  a  notch 
in  the 
stern  ; 
in  this 
notch 
put  a 

piece  of  camphor  gum  (Fig.  19).  The  camphor 
will  weaken  the  tension  astern,  while  the  tension 
at  the  bow  will  draw  the  boat  forward. 

Surface  tension  makes  a  soap  bubble  contract. 
Blow  a  bubble  on  a  small  funnel  and  hold  the 
open  tube  near  a  candle  flame  (Fig.  20).  The  ex-  Fig.  20 


SURFACE  PHENOMENA 


pelled  air  will  blow  the  flame  aside,  and  the  smaller  the  bubble  the 
more  energetically  will  it  expel  the  air. 

A  small  cylinder  of  fine  wire  gauze  with  solid  ends,  if  completely 
immersed  in  water  and  partly  filled,  may  be  lifted  out  horizontally 
and  hold  the  water.  A  film  fills  the  meshes  of  the  gauze  and  makes 
the  cylinder  air-tight ;  if  it  is  broken  by  blowing  sharply  on  it,  the 
water  will  quickly  run  out. 

32.  Capillary  Elevation  and  Depression.  —  If  a  fine  glass 
tube,  commonly  called  a  capillary  or  hairlike  tube,  is 
partly  immersed  vertically  in  water,  the 
water  will  rise  higher  in  the  tube  than  the 
level  outside  ;  on  the  other  hand,  mercury 
is  depressed  below  the  level  outside.  The 
top  of  the  little  column  of  water  is  concave, 
while  that  of  the  column  of  mercury  is 
convex  upward  (Fig.  21). 

Familiar  examples  of  capillary  action  are  numer-  pig.  21 

ous.  Blotting  paper  absorbs  ink  in  its  fine  pores, 
and  oil  rises  in  a  wick  by  capillary  action.  A  sponge  absorbs  water 
for  the  same  reason ;  so  also  does  a  lump  of  sugar.  A  cotton  or  a 
hemp  rope  absorbs  water,  increases  in  diameter,  and  shortens.  A 
liquid  may  be  carried  over  the  top  of  a  vessel  by  capillary  action  in  a 
large  loose  cord,  like  water  in  a  siphon.  Many  salt  solutions  con- 
struct their  own  capillary  highway  up  over  the  top  of  the  open  glass 
vessel  in  which  they  stand.  They  first  rise  by  capillary  action  along 
the  surface  of  the  glass,  then  the  water 
evaporates,  leaving  the  salt  in  fine  crystals, 
through  which  the  solution  rises  by  capil- 
lary action  still  higher.  This  process  may 
continue  until  the  liquid  flows  over  the  top 
and  down  the  outside  of  the  vessel. 


33.    Laws   of    Capillary   Action. — 

Support  vertically  several  clean  glass  tubes 
of  small  internal  diameter  in  a  vessel  of 
pure  water  (Fig.  22).  The  water  will  rise 
in  these  tubes,  highest  in  the  one  of  small- 


Fig.  22 


24 


MOLECULAR  PHYSICS 


est  diameter,  and  least  in  the  one  of  greatest.  With  mercury  in 
place  of  water,  the  depression  will  be  the  greatest  in  the  smallest  tube. 
If  two  chemically  clean  glass  plates,  inclined  at  a  very  small 
angle,  be  supported  with  their  lower  edges  in  water,  the  height  to 
which  the  water  will  rise  at  different  points  will  be  inversely  as 

the  distance  between  the  plates, 
and  the  water  line  will  be  curved 
as  in  Fig.  23. 


These  experiments  illus- 
trate the  following  laws  : 

I.  Liquids  ascend  in 
tubes  when  they  wet  them, 
that  is,  ^vhen  the  surface 

is  concave;  and  they  are   depressed  when  they  do  not 
wet  them,  that  is,  when  the  surface  is  convex. 

II.    For  tubes  of  sTnall  diameter,  the  elevation  or  de- 
pression is  inversely  as  the  diameter  of  the  tube. 

34.  Capillary  Action  in  Soils.  —  The  distribution  of  mois- 
ture in  the  soil  is  greatly  affected  by  capillarity.     Water 
spreads  through  compact  porous  soil  as  tea  spreads  through 
a  lump  of  loaf  sugar.     As  the  moisture  evaporates  at  the 
surface,  more  of  it  rises  by  capillary  action  from  the  sup- 
ply below.     To    conserve   the   moisture  in   dry   weather 
and  in  "  dry  farming,"  the  surface  of  the  soil  is  loosened 
by  cultivation,  so  that  the   interstices  may  be  too  large 
for  free  capillary   action.     The    moisture    then   remains 
at   a   lower    level,   where   it   is   needed    for  the  growth 
of  plants. 

35.  Capillarity  related  to  Surface  Tension.  —  The  attrac- 
tion of  glass  for  water  is  greater  than  the  attraction  of 
water  for  itself  (§  10).     When  a  liquid  is  thus  attracted 
by  a  solid,  the  liquid  wets  it  and  rises  with  a  concave 


MOLECULAR  FORCES  IN  SOLIDS  25 

surface   upward    (Fig.    24).      The   surface    tension  in  a 
curved  film  makes  the  film  contract  and  produces  a  pres- 
sure towards  its  center   of   curvature,   as   shown  in  the 
case   of   the   soap  bubble  (§    31).     When 
the  surface    of   the  liquid   in  the  tube   is 
concave,  the  resultant  of  this  normal  pres- 
sure is   a   force   upward;    the    downward 
pressure    of    the   liquid  under  the  film  is 
thus  reduced,  and   the    liquid   rises  until 
the  weight  of  the  column  AE  downward 
just  equals  the    resultant   of   the    normal 
forces    of    the    film    upward.      When   the 
liquid  does  not  wet  the  tube,  the  normal 
pressure    of    the    film   is    downward,    and 
the  column  sinks  until  the  downward  pressure  is  counter- 
balanced by  the  upward  pressure  of  the  liquid  outside. 

III.    MOLECULAR  FORCES  IN  SOLIDS 

36.  Solution  of  Solids.  —  The  solution  of  certain  solids 
in  liquids  has  become  familiar  by  the  use  of  salt  and  sugar 
in  liquid  foods.  The  solubility  of  solids  is  limited,  for  it 
depends  on  the  nature  of  both  the  solid  and  the  solvent,  — 
the  liquid  in  which  it  dissolves.  At  room  temperatures, 
table  salt  dissolves  about  three  times  as  freely  in  water  as 
in  alcohol ;  while  grease,  which  is  practically  insoluble  in 
water,  dissolves  readily  in  benzine  or  gasoline. 

Solution  in  a  small  degree  takes  place  in  many  unsuspected  cases. 
Thus,  certain  kinds  of  glass  dissolve  to  an  appreciable  extent  in  hot 
water.  Many  rocks  are  slightly  soluble  in  water,  and  the  familiar 
adage  that  the  "  constant  dropping  of  water  wears  away  a  stone  "  is 
accounted  for,  in  part  at  least,  by  the  solution  of  the  stone.  Flint 
glass,  out  of  which  cut  glass  vessels  are  made,  dissolves  to  some  extent 
in  aqua  ammonia ;  this  liquid  should  not  be  kept  in  cut  glass  bottles, 
nor  should  cut  glass  be  washed  in  water  containing  ammonia. 


26  MOLECULAR   PHYSICS 

There  is  a  definite  limit  to  the  quantity  of  a  solid  which 
will  dissolve  at  any  temperature  in  a  given  volume  of  a 
liquid.  For  example,  360  gm.  of  table  salt  will  dissolve 
in  a  liter  of  water  at  ordinary  temperatures ;  this  is  equiva- 
lent to  three  quarters  of  a  pound  to  the  quart.  When  the 
solution  will  dissolve  no  more  of  the  solid,  it  is  said  to  be 
saturated.  As  a  general  rule,  though  it  is  not  without 
exceptions,  the  higher  the  temperature,  the  larger  the 
quantity  of  a  solid  dissolved  by  a  liquid.  A  liquid  which 
is  saturated  at  a  higher  temperature  is  supersaturated  when 
cooled  to  a  lower  one. 

37.  Crystallization.  —  When  a  saturated  solution  evapo- 
rates, the  liquid  only  passes  off  as  a  vapor ;  the  dissolved 
substance  remains  behind  as  a  solid.  When  the  solid 
thus  separates  slowly  from  the  liquid  and  the  solution 
remains  undisturbed,  the  conditions  are  favorable  for 
the  molecules  to  unite  under  the  influence  of  their 
mutual  attractions,  and  they  assume  regular  geometric 
forms  called  crystals.  Similar  conditions  exist  when 
a  saturated  solution  cools  and  becomes  supersaturated. 
The  presence  of  a  minute  crystal  of  the  solid  then  in- 
sures the  formation  of  more.  The  process  of  the  sep- 
aration of  a  solid  in  the  form  of  crystals  is  known  as 
crystallization. 

Dissolve  100  gm.  of  common  alum  in  a  liter  of  hot  water.  Hang 
some  strings  in  the  solution  and  set  aside  in  a  quiet  place  for  several 
hours.  The  strings  will  be  covered  with  beautiful  transparent  octa- 
hedral crystals.  Copper  sulphate  may  be  used  in  place  of  the  alum ; 
large  blue  crystals  will  then  collect  on  the  strings. 

Filter  a  saturated  solution  of  common  salt  and  set  aside  for  twen- 
ty-four hours.  An  examination  of  the  surface  will  reveal  groups  of 
crystals  floating  about.  Each  one  of  these,  when  viewed  through  a 
magnifying  glass,  will  be  found  to  be  a  little  cube. 

Ice  is  a  compact  mass  of  crystals,  and  snow  consists  of  crystals 


MOLECULAR  FORCES  IN  SOLIDS 


27 


formed  from  the  vapor  of  water.     They  are  of  various  forms,  but  all 
hexagonal  in  outline  (Fig.  25).1 

38.  Molecular  Strength.  —  To  tear  apart  a  solid,  a 
stretching  force  must  be  applied  in  excess  of  the  forces 
holding  together  the  molecules  on  opposite  sides  of  the 


Fig.  25 

fracture.     Tenacity  (§  12)  is  accounted  for  by  this  force 
of  cohesion  and  is  a  measure  of  it. 

Steel  has  the  greatest  tensile  strength  of  all  metals ;  a 
steel  rod  1  sq.  in.  in  cross  section  requires  a  stretching 
force  of  about  65  tons  to  break  it.  For  the  same  cross 
section,  hard  drawn  copper  breaks  under  a  tension  of  from 
23  to  34  tons.  The  breaking  tension  varies  as  the  cross 
section. 

39.  Molecular  Equilibrium.  —  When  a  semiliquid  solid, 
such  as  glass  at  a  high  temperature,  is  suddenly  chilled, 
the  molecules  do  not  have  time  to  arrange 
themselves  under  the  cohesive  forces  acting 
on  them,  and  the  molecular  grouping  is  one 
of  more  or  less  unstable  restraint. 

Prince  Rupert  drops  (Fig.  26)  are  made  by  dropping 
melted  glass  into  cold  water.     The  outside  is  suddenly 
chilled  and  solidified,  while  the  interior  is  still  fused,  and  when  it 
cools  it  must  accommodate  itself  to  the  dimensions  of  the  outer  skin. 


Fig.  26 


1  These  figures  were  made  from   microphotographs  taken  by  Mr.  W.  A, 
Bentley,  Jericho,  Vt. 


28  MOLECULAR  PHYSICS 

The  drop  is  thus  under  great  tension.  With  a  pair  of  pliers  break 
off  the  tip  of  the  drop  under  water  in  a  tumbler,  or  scratch  with  a 
file ;  the  whole  drop  will  fly  to  powder  with  almost  explosive  violence. 
A  large  tall  jar  on  foot  is  usually  thick  at  the  bottom,  and  has  been 
imperfectly  annealed.  Such  jars  have  not  infrequently  been  broken 
by  a  scratch  inside,  made,  for  example,  by  stirring  emery  powder  in 
water  by  means  of  a  long  wooden  stick.  A  scratch  inside  is  usually 
fatal  to  a  lamp  chimney. 

40.  Elasticity.  —  Apply  pressure  to  a  tennis  ball,  stretch 
a  rubber  band,  bend  a  piece  of  watch  spring,  twist  a  strip 
of  whalebone.     In  each  case  the  form  or  the  volume  has 
been  changed,  and  the  body  has  been  strained.     A  strain 
means  either  a  change  of  size  or  a  change  of  shape.     As 
soon  as  the  distorting  force,  or  stress,  has  been  withdrawn, 
these  bodies  recover  their  initial  volume  and  dimensions. 
This  property  of  recovery  from  a  strain  when  the  stress  is 

.removed  is  called  elasticity.  It  is  called  elasticity  of  form 
when  a  body  recovers  its  form  after  distortion;  and 
elasticity  of  volume  when  the  temporary  distortion  is  one 
of  volume.  Gases  and  liquids  have  perfect  elasticity  of 
volume,  because  they  recover  their  former  volume  when 
the  original  pressure  is  restored.  They  have  no  elasticity 
of  form.  Some  solids,  such  as  shoemaker's  wax,  putty, 
and  dough,  when  long-continued  force  is  applied,  yield 
slowly  and  never  recover. 

The  elasticity  of  a  body  may  be  called  forth  by  pressure, 
by  stretching,  by  bending,  or  by  twisting.  The  bound- 
ing ball  and  the  popgun  are  illustrations  of  the  first; 
rubber  bands  are  familiar  examples  of  the  second  ;  bows 
and  springs  of  the  third ;  and  the  stretched  spiral  spring 
exemplifies  the  fourth. 

41.  Hooke's  Law.  —  Solids  have  a  limit  to  their  distor- 
tion, called  the  elastic  limit,  beyond  which  they  yield  and 
are  incapable  of  recovering  their  form  or  volume.     The 


QUESTIONS  AND  PROBLEMS  29 

elastic  limit  of  steel  is  very  high ;  steel  breaks  before  there 
is  much  permanent  distortion.  On  the  other  hand,  lead 
does  not  recover  completely  from  any  distortion. 

When  the  strain  in  an  elastic  body  does  not  exceed  the 
elastic  limit,  in  general  the  distortion  is  proportional  to  the 
distorting  force.  This  relation  is  known  as  Hooke's  law. 

Clamp  a  meter  stick  to  a  suitable  support  (Fig.  27),  and  load  the 
free  end  with  some  convenient  weight  in  a  light  scale  pan ;  observe 
the  bending  of  the 
stick  by  means  of 
the  vertical  scale  and 
the  pointer.  Then 
double  the  weight 
and  note  the  new 
deflection.  It  should 

be  double  the   first.  P. 

The  amount  of  bend- 
ing or  distortion  of  the  bar  is  proportional  to  the  weight. 

Generally,  for  all  elastic  displacements  within  the 
elastic  limit,  the  distortions  of  any  kind,  due  to  bending, 
stretching,  or  twisting,  are  proportional  to  the  forces  pro- 
ducing them. 

Questions  and  Problems 

1.  When  a  glass  tube  is  cut  off  its  edges  are  sharp ;  why  do  they 
become  rounded  by  softening  in  a  blowpipe  flame? 

2.  Why  does  a  small  vertical  stream  of  water  break  into  drops  ? 

3.  Why  can  you  not  write  with  ink  on  blotting  paper  ? 

4.  Explain  the  action  of  blotting  paper ;  of  a  towel  in  drying  the 
wet  hands  ;  of  a  sp'onge. 

5.  A  soap  bubble  is  filled  with  air.     Is  the  air  inside  denser  or 
rarer  than  the  air  outside  ? 

6.  If  dry  wooden  wedges  are  driven  into  holes  or  a  channel  in  a 
stone,  and  are  then  wet  with  water,  what  is  the  effect  ? 

7.  Why  does  an  automobile  ride  easier  with  pneumatic  tires  than 
with  solid  rubber  ones  ? 


30  MOLECULAR  PHYSICS 

8.  Are  the  divisions  on  the  scale  of   a  spring  balance  equal? 
What  law  is  illustrated? 

9.  If  an  iron  wire  one  tenth  inch  in  diameter  will  safely  support 
300  pounds,  how  many  pounds  will  one  support  one  fifth  inch  in 
diameter  ? 

10.  If  a  steel  rod  10  ft.  long  is  stretched  within  its  elastic  limit 
0.15  in.  by  a  certain  weight,  how  much  would  a  rod  20  ft.  long  be 
lengthened  by  the  same  weight  ?  By  half  the  weight? 


CHAPTER   III 

MECHANICS  OF  FLUIDS 

I.     PRESSURE  OF  FLUIDS 

42.  Characteristics  of  Fluids.  —  A  fluid  has  no  shape  of 
its  own,  but  takes  the  shape  of  the  containing  vessel.     It 
cannot  resist  a  stress  unless  it  is  supported  on  all  sides. 
The  molecules  of  a  fluid  at  rest  are  displaced  by  the  slight- 
est force ;  that  is,  a  fluid  yields  to  the  continued  applica- 
tion of  a  force  tending  to  change  its  shape.     But  fluids 
exhibit  wide  differences  in  mobility,  or  readiness  in. yield- 
ing to  a  stress.     Alcohol,  gasoline,  and  sulphuric  ether 
are  examples  of  very  mobile  liquids;    glycerine  is  very 
much  less  mobile,  and  tar  still  less  so.    'In  fact,  liquids 
shade  off  gradually  into  solids.     A-  stick*  of,  sealing  wax 
supported   at   its   ends   yields   continuously   to   its   own 
weight;  in  warm  weather  paraffin  candles  do  not  main- 1 
tain  an  upright  position  in  a  candlestick,  but  curve  over 
or  bend  double;    a  cake  of  shoemaker's  wax  on  water, 
with  bullets  on  it  and  corks  under  it,  yields  to  both  alid 
is  traversed  by  both  in  opposite  directions.     It  will  even' 
flow  very  slowly  down  a  tortuous  channel.     At  the  same 
time,  sealing  wax  and  shoemaker's  wax  when  cold  break 
readily  under  the  blow  of  a  hammer. 

43.  Viscosity.  —  The  resistance  of  a  fluid  to  flowing  under 
stress  is  called  viscosity.     It  is  due  to  molecular  friction. 
The  slowness  with  which  a  fine  precipitate,  thrown  down 
by  chemical  action,  settles  in  water  is  owing  to  the  vis- 

31 


32  MECHANICS   OF  FLUIDS 

cosity  of  the  liquid;  and  the  slow  descent  of  a  cloud  is 
accounted  for  by  the  viscosity  of  the  air.  Viscosity  varies 
between  wide  limits.  It  is  less  in  gases  than  in  liquids; 
hot  water  is  less  viscous  than  cold  water ;  hence  the  rela- 
tive ease  with  which  a  hot  solution  filters. 

44.  Liquids  and  Gases.  —  Fluids  are  divided  into  liquids 
and  gases.     Liquids,  such  as  water  arid  mercury,  are  but 
slightly  compressible,  while  gases,  such  as  air  and  hydro- 
gen, are  highly  compressible.     A  liquid  offers  great  resist- 
ance to  forces  tending  to  diminish  its  volume,  while  a  gas 
offers  relatively  small  resistance.      Water  is  reduced  only 
0.00005  of  its  volume  by  a  pressure  equal  to  that  of  the 
atmosphere  (practically  15  Ib.  to  the  square  inch),  while 
air  is  reduced  to  one  half  its  volume  by  the  same  additional 
pressure. 

Then,  too,  gases  are  distinguished  from  liquids  by  the 
fact  that  any  mass  of  gas  when  introduced  into  a  closed 
vessel  always  completely  fills  it,  whatever  its  volume.  A 
liquid  has  a  bulk  of  its  own,  but  a  gas  has  not,  since  a 
gas  expands  indefinitely  as  the  pressure  on  it  decreases. 

45,  Pascal's  Principle.  —  A  solid  transmits  pressure  only 
in  the  direction  in  which  the  force  acts ;  but  a  fluid  trans- 
mits pressure  in  every  direction.     Hence  the  law: 

Pressure  applied  to  an  inclosed  fluid  is  transmitted 
equally  in  all  directions  and  without  diminution  to 
every  part  of  the  fluid  and  of  the  interior  of  the  con- 
taining vessel. 

This  is  the  fundamental  law  of  the  mechanics  of  fluids. 
It  is  a  direct  consequence  of  their  mobility,  and  it  applies 
to  both  liquids  and  gases.  It  was  first  announced  by 
Pascal  in  1653. 


Galileo  Galilei  (1566- 
1642)  was  born  at  Pisa,  Italy. 
He  was  a  man  of  great  gen- 
ius, and  an  experimental 
philosopher  of  the  first  rank. 
He  was  educated  as  a  phys- 
ician, but  devoted  his  life  to 
mathematics  and  physics. 
He  discovered  the  properties 
of  the  pendulum,  invented 
the  telescope  bearing  his 
name,  and  was  ardent  in  his 
support  of  the  doctrine  that 
the  earth  revolves  around 
the  sun.  Besides  his  original 
work  in  physics,  he  made  interesting  discoveries  in  astronomy. 


Blaise  Pascal  (1623-1662) 
was  born  at  Clermont  in  Au- 
vergne.  He  was  both  a  math- 
ematician and  a  physicist. 
Even  as  a  youth  he  showed 
remarkable  learning,  and  at 
the  age  of  seventeen  achieved 
renown  with  a  treatise  on 
conic  sections.  He  is  best 
known  for  his  announcement 
in  1658  of  the  important  law 
of  fluid  pressure  bearing  his 
name.  He  distinguished  him- 
self by  his  researches  in  conic 
sections,  in  the  properties  of 
the  cycloid,  and  the  pressure  of  the  atmosphere. 


PRESSURE  OF  FLUIDS 


33 


Fig.  28 


ILLUSTRATIONS.  —  Fit  a  perforated  stopper  to  an  ounce  bottle,  pref- 
erably with  flat  sides,  and  mounted  in  a  suitable  frame  (Fig.  28). 
Fill  the  bottle  with  water  and  then  force  a  metal 
plunger  through  the  hole  in  the  stopper.  If  the 
plunger  fits  the  stopper  water-tight,  the  pressure  ap- 
plied to  the  plunger  will  be  transmitted  to  the  water 
as  a  bursting  pressure;  and  the  whole  pressure  trans- 
mitted to  the  inner  surface  of  the  bottle  will  be  as 
much  greater  than  the  pressure  applied  as  the  area 
of  this  surface  is  greater  than  that  of  the  end  of  the 
plunger. 

Figure  29  is  a  form  of  syringe  made  of  glass ;  the 
hollow  sphere  at  the  end  has  several  small  openings. 
Fill  with  water  and  apply  pres- 
sure to  the  piston.  The  water 
will  escape  in  a  series  of  jets  of  apparently  equal 
velocities,  although  only  one  of  them  is  directly 
in  line  with  the  piston. 

Fit  a  glass  tube  to  the  stem  of  a  small  rubber 
balloon;  blow  into  the  tube;  the  balloon  will 
expand  equally  in  all  directions,  forming  a  sphere 
and  showing  equal  pressures  in  all  directions. 
A  large  soap  bubble  shows  the  same  thing. 

46.    The  'Hydraulic  Press.  —  An  impor- 
tant application  of  Pascal's  principle  is 
the  hydraulic  press.    Figure  30  is  a  section 
showing  the  prin- 
A    heavy    piston    P 

works  water-tight  in  the  larger  cyl- 
inder A,  while  in  the  smaller  one 

the  piston  p  is  moved  up  and  down  A 

as  a  force  pump ;    it  pumps  water 

or   oil  from   the   reservoir  D  and 

forces  it  through  the  tube  0  into 

the  cylinder  A.     When  the  piston 

p  of  the  pump  is  forced  down,  the  Fig.  30 


Fig.  29 

cipal   parts. 


34 


MECHANICS   OF  FLUIDS 


liquid  transmits  the  pressure  to  the  base  of  the  larger 
piston,  on  which  the  force  R  is  as  many  times  the  force  E 
applied  to  p  as  the  area  of  the  large  piston  is  greater  than 
the  area  of  the  small  one.  If  the  cross  sectional  area  of 
the  small  piston  is  represented  by  #,  and  that  of  the  large 
one  by  A,  the  ratio  between  the  forces  acting  on  the  two 
pistons  is 

JR     A  =  1P 

E      a       #' 

where  D  and  d  are  the  diameters  of  the  large  and  small 

pistons  respectively. 

Figure  31  illustrates  a  press  driven  by  a  belt  on  the  pulley 

P.     F  and  F  are  pumps  forcing  water  into  the  cylinder  B. 

In  the  hydraulic  press 
it  is  evident  that  the 
small  piston  travels  as 
many  times  farther  than 
the  large  one  as  the  force 
exerted  by  the  large  pis- 
ton is  greater  than  the 
effort  applied  to  the  small 
one. 

47.  The  Hydraulic  Ele- 
vator. —  A  modern  appli- 
cation of  Pascal's  prin- 
ciple is  the  hydraulic 
elevator.  A  simple  form 
is  shown  in  Fig.  32.  A 
long  piston  P  carries  the 
cage  A,  which  runs  up 


Fig.  31 


and  down  between  guides  and  is  partly  counterbalanced  by 
a  weight  W.     The  piston  runs  in  a  tube  or  pit  sunk  to  a 


PRESSURE  OF  FLUIDS 


35 


depth  equal  to  the  height  to  which  the  cage  is  designed  to 
rise.  Water  under  pressure  enters  the  pit  from  the  pipe 
m  through  the  valve  v.  Turned  in  one  /\-?\ 

,~,       •.--><—! 

direction  the  valve  admits  water  to  the 
sunken  cylinder,  and  the  pressure  forces 
the  piston  up  ;  turned  in  the  other  direc- 
tion it  allows  the  water  to  escape  into 
the  sewer,  and  the  elevator  descends  by 
its  own  weight. 

When  greater  speed  is  required,  the  cage 
is  connected  to  the  piston  indirectly  by  a 
system  of  pulleys.  The  cage  then  usually 
runs  four  times  as  fast  as  the  piston. 

48.    Downward  Pressure  of  a  Liquid.  — 

The  weight  of  each  layer  of  a  liquid  is 
transmitted  to  every  layer  at  a  lower 
level. 


A  glass   cylinder  is  cemented  into   a  metal 
ferule  which  screws  into  a  second  short  cylinder, 
across  the  base  of  which  is  tied 
a  disk   of   sheet   rubber.      The 
pointer  below  acts   as   a  lever, 
the  short  arm  pressing  against 
the  center  of   the  rubber  disk, 
and  the  long  arm  moving  over  a 
scale  (Fig.  33).     It  should  be  adjusted  to  point  to 
zero   to  begin  with.     Fill  the 
glass  cylinder  one  third  full  of 
water  and  note  the  reading  of 
the  pointer  on  the  scale.     Add 
water  until  the  cylinder  is  two 
Fig.  33  thirds  full ;  the  reading  of  the 

pointer  will  be  doubled.  Fi- 
nally fill  the  cylinder,  and  the  reading  on  the  scale  should  be  three 
times  the  first  one.  Hence, 


MECHANICS  OF  FLUIDS 


The  downward  pressure  of  a  liquid  is  proportional  to 
the  depth. 

Repeat  the  experiment  with  a  saturated  solution  of  common  salt, 
wliirh  is  heavier  than  water.  Every  pointer  reading  will  be  greater 
than  the  corresponding  ones  with  water,  but  the  same  relation  will 
exist  between  them.  Hence, 

The  downward  pressure  of  a  liquid  is  proportional  to 
its  density  (§59). 

49.  Pressure  at  a  Point.  —  The  three  glass  tubes  of  Fig. 
34  have  short  arms  of  the  same  length,  measured  from  the 

bend  to  the  mouth.  They  open  in 
different  directions,  —  upward,  down- 
ward, and  sidewise.  Place  mercury 
to  the  same  depth  in  all  the  tubes,  and 
lower  them  into  a  tall  jar  filled  with 
water.  When  the  open  ends  of  the 
short  arms  are  kept  at  the  same  level, 
the  change  in  the  level  of  the  mercury 
is  the  same  in  all  of  them.  Hence, 

The  pressure  at  a  point  in  a  liquid 

is  the  same  in  all  directions. 

The  equality  of  -pressure  in  all  di- 

rections may  also  be  inferred  from  the 

absence  of 

currents 

in  ;i  vessel 

of  liquid, 

since   ;m 

u  n  b  a  1  - 

anced  pressure  would  produce 
motion  of  the  liquid.  Fig'  35 

50.  Pressure  Independent  of  the  Shape  of  the  Vessel.  - 
Proceeding  as  in  §  48,  use  in  succession  the  three  vessels 


-  34 


PRESSURE  OF  FLUIDS  37 

shown  in  Fig.  35.  They  have  equul  bases,  but  differ  in 
shape  and  volume.  They  are  known  as  Pascal's  vases. 
Fill  each  in  succession  to  the  same  height,  and  note  the 
reading  of  the  pointer.  It  will  be  the  same  for  all,  not- 
withstanding the  great  difference  in  the  amount  of  water. 
Hence, 

The  downward  pressure  in  a  liquid  is  independent  of  ^ 
the  shape  of  the  vessel. 

The  apparent  contradiction  of  unequal  masses  of  a  liquid 
producing  equal  pressures  is  known  as  the  hydrostatic  para- 
dox. It  is  only  another  form  of  Pascal's  principle. 

51.  Total  Pressure  on  Any  Surface.  —  The  total  pressure 
of  a  liquid  on  any  horizontal  surface  is  equal  to  the  weight 
of  a  column  of  the  liquid  whose  base  is  the  area  pressed  uxwn, 
and  whose  height  is  the  depth  of  this  area  below  the  surface 
of  the  liquid.  /) 

Let  A  denote  the  area  pressed  upon,  H  its  depth,  and     I 
d  the  weight  of  a  unit  volume  of  the  liquid.      Then  the 
whole  pressure  on  this  area  is 

P^AHd.     .     .     .    (Equation  1) 

The  pressure  on  any  immersed  surface  of  any  inclina- 
tion or  shape  is  found  by  computing  the  pressures  on  all 
the  elementary  areas  into  which  the  surface  may  be  divided 
and  adding  them  together.  The  result  is  expressed  as 
follows  : 

The  total  pressure  on  any  immersed  surface  is  equal  to  the 
weight  of  a  column  of  the  liquid  whose  base  has  an  area  equal 
to  that  of  the  surface  pressed  upon,  and  whose  height  is  equal 
to  the  depth  of  the  center  of  gravity  of  this  surface  below  the 
surface  of  the  liquid. 

Formula  (1)  applies  to  both  cases.     In  the  English  sys- 


38  MECHANICS  OF  FLUIDS 

tern  d  for  water  is  62.4  Ib.  per  cubic  foot;  in  the  metric 
system  it  is  1  gm.  per  cubic  centimeter. 

EXAMPLE.  The  upstream  face  of  a  dam  measures  20  ft.  from  top 
to  bottom,  but  it  slopes  so  that  its  center  of  figure  is  only  7  ft.  from 
the  surface  of  the  water  when  the  dam  is  full.  Find  the  perpendicu- 
lar pressure  against  the  dam  for  every  foot  of  length. 

SOLUTION.  The  area  of  the  face  of  the  dam  per  foot  in  length  is 
*20  sq.  ft.  Hence  the  weight  of  the  column  of  water  to  represent  the 
pressure  is  20  x  7  x  62.4  =  8736  Ib. 

52.  Surface  of  a  Liquid  at  Rest.  —  The  free  surface  of  a 
liquid  under  the  influence  of  gravity  alone  is  horizontal. 
Even  viscous  liquids  assume  a  horizontal  surface  in  course 
of  time.  The  sea,  or  any  other  large  expanse  of  water,  is 
a  part  of  the  spheroidal  surface  of  the  earth.  When  one 
looks  with  a  field  glass  at  a  long  straight  stretch  of  the 

Suez  Canal,  the  water  and  the  re- 
taining wall  as  contrasting  bodies 
appear  distinctly  curved  as  a  por- 
tion of  the  rounded  surface  of  the 
earth. 


53.  Level  of  Liquid  in  Connected 
Vessels.  —  The  water  in  the  ap- 
paratus of  Fig.  36  rises  to  the  same 
level  in  all  the  branches.  There  is 
equilibrium  because  the  pressures 
on  opposite  sides  of  any  cross  sec- 
Fi&-  36  tion  of  the  liquid  in  the  connect- 

ing tube  are  equal,  since  they  are  due  to  liquid  columns  of 
the  same  height. 

The  glass  water  gauge,  used  to  show  the  height  of  the  water  in  a 
steam  boiler,  is  an  application  of  this  principle ;  also  the  water  level, 
consisting  of  two  glass  tubes,  joined  by  a  long  rubber  tube,  and  em- 
ployed by  builders  for  leveling  foundations. 

Artesian  or  flowing  wells  illustrate  on  a  grand  scale  the  tendency  of 


QUESTIONS  AND  PROBLEMS  39 

water  "  to  seek  its  level."  In  geology  an  artesian  basin  is  one  com- 
posed of  superposed  strata  of  great  extent,  one  of  which  is  permeable 
to  water,  and  lies  between  two  of  clay  or  other  material  through  which 
water  does  not  percolate  (Fig.  37) .  This  stratum  crops  out  at  some 
A 


Fig.  37 

higher  level  where  water  finds  entrance,  as  at  A.  When  a  well  is 
bored  through  the  overlying  strata  in  the  valley,  water  issues  on  ac- 
count of  the  pressure  transmitted  from  higher  points  at  a  distance. 
There  are  8000  or  10,000  artesian  wells  in  the  western  part  of  the 
United  States;  some  notable  ones  are  at  Chicago,  St.  Louis,  New 
Orleans,  Charleston,  and  Denver.  In  Europe  there  are  very  deep 
flowing  wells  in  Paris  (2360  ft.),  Berlin  (4194  ft.),  and  near  Leipzig 
(5740  ft). 

Questions  and  Problems 

1.  Why   do    gas   bubbles   rising  through  water  from  a  marshy 
bottom  grow  larger  as  they  ascend? 

2.  Why  is  the  water  pressure  greater  in  the  basement  of  a  house 
than  on  the  top  floor  ? 

3.  A   force  of  150  Ib.  is  applied  to  a  small  piston   of   an  hy- 
draulic press;  the  two  pistons  have  diameters  of  1  in.  and  5  in.  respec- 
tively ;  what  pressure  is  exerted  on  the  larger  one  ? 

4.  A  swimming  tank  50  ft.  square  is  filled  with  water  to  a  depth 
of  10  ft.     What  is  the  total  pressure  on  the  bottom  ?    On  one  side  ? 

5.  A  glass  cylinder  76  cm.  high  is  level  full  of  mercury.     What  is 
the  pressure  in  grams  per  square  centimeter  on  the  bottom  ?     (1  cm.3 
of  mercury  weighs  13.6  gm.) 

6.  A  recording  pressure  gauge  registered  zero  at  the  surface  of  a 
fresh-water  lake  and  150  Ib.  per  square  inch  at  the  bottom.     Calculate 
the  depth  of  the  lake. 

7.  How  high  would  water  rise  in  the  pipes  of  a  building  if  a 
pressure  gauge  shows  that  the  pressure  at  the  ground  floor  is  40  Ib. 
per  square  inch.  ? 


40  MECHANICS   OF  FLUIDS 

8.  A  cylindrical  steel  tank  has  an  internal  diameter  of  20  ft.  and  a 
height  of  25  ft.     When  it  is  filled  with  kerosene,  weight  56  Ib.  per  cubic 
foot,  what  is  the  total  pressure  on  the  bottom  ?     On  the  cylindrical  side  ? 

9.  An  hydraulic  lift  carries  an  unbalanced  weight  of  3000  Ib.     If 
the  piston  supporting  the  lift  is  8  in.  in  diameter,  what  pressure  of 
water  per  square  inch  will  be  necessary  ? 

10.  A  vertical  tube  is  filled  with  mercury,  weighing  13.6  gm.  per 
cubic  centimeter,  to  a  depth  of  3  m.     What  is  the  pressure  in  grams 
per  square  centimeter  on  the  bottom  ? 

11.  What  is  the  pressure  per  square  foot  at  a  depth  of  2  mi.  in  the 
ocean,  sea  water  weighing  64  Ib.  per  cubic  foot? 

12.  A  diver  is  working  at  a  depth  of  30  ft.     What  is  the  pressure 
per  square  inch  on  the  surf  ace  of  his  body  ?    In  fresh  water  ?    In  the 
ocean  ? 

13.  A  hole  in  the  bottom  of  a  ship  25  ft.  below  the  surface  of  the 
water  is  covered  with  canvas.     What  is  the  pressure  per  square  inch 
against  the  canvas? 

14.  An  oak  cask  2  ft.  high  stands  on  end  and  into  its  head  is 
screwed  a  vertical  iron  pipe  an  inch  in  diameter  and  29  ft.  high.  The 
cask  and  the  pipe  are  both  filled  with  water  to  the  top.  The  cask  has 
a  bunghole  midway  between  its  ends  and  2  in.  in  diameter.  What  is 
the  total  pressure  on  the  bung? 

II.     BODIES  IMMERSED  IN  LIQUIDS 

54.  Buoyancy.  —  A  fresh  egg  sinks  in  fresh  water  and 
floats  in  brine  ;  a  marble  sinks  in  wat.er  and  floats  in  mer- 
cury; a  piece  of  oak  floats  in  water,  but  a  piece  of  the 
dense  wood  known  as  "lignum  vitas"  sinks.  When  a 
bather  wades  up  to  his  neck  in  the  sea,  he  is  nearly  lifted 
off  his  feet  by  the  buoyant  force  of  the  water.  These 
facts  show  that  the  resultant  pressure  of  a  liquid  on  a 
body  immersed  in  it  is  a  vertical  force  upward,  and  that 
it  counterbalances  a  part  or  the  whole  of  a  body's  weight. 
The  resultant  upward  pressure  of  a  liquid  on  a  body  immersed 
in  it  is  called  buoyancy . 


BODIES  IMMERSED  IN  LIQUIDS 


41 


55.  Archimedes'  Principle.  —  Archimedes  discovered  the 
law  of  buoyancy  about  240  B.C.  while  attempting  to  de- 
termine the  composition  of  the  golden  crown  of  Hiero  II, 
King  of  Syracuse,  who  suspected  that  the  goldsmith  had 
mixed  base  metal  with  the  gold.  The  law  is: 

A  body  immersed  in  a  liquid  is  buoyed  up  by  a  force 
equal  to  the  weight  of  the  liquid  displaced  by  it. 

If  a  cube  be  immersed  in  water  (Fig.  38),  the  pressures 

on  the  vertical  sides  a  and  b  are  equal  and  in  opposite 

directions.     The  same  is  true  of 

the  other  pair  of  vertical  faces. 

There  is  therefore  no  resultant 

horizontal  pressure.     On  d  there 

is  a  downward  pressure  equal  to 

the  weight  of  the  column  of  water 

having  the  face  d  as  a  base,  and 

the  height  dn.     On  c  there  is  an 

upward    pressure    equal    to    the 

weight    of   a    column    of    water 

whose  base  is  the  area  of  0,  and  whose  height  is  en.  The 
upward  pressure  therefore  exceeds  the 
downward  pressure  by  the  weight  of  the 
prism  of  water  whose  base  is  the  face  c  of 
the  cube,  and  whose  height  is  the  difference 
between  Qn  and  dn,  or  cd.  This  is  the 
weight  of  the  liquid  displaced  by  the  cube. 

A  metallic  cylinder  5.1  cm.  long,  and  2.5  cm.  in 
diameter  has  a  volume  of  almost  exactly  25  cm.3. 
Suspend  it  by  a  fine  thread  from  one  arm  of  a  bal- 
ance (Fig.  39)  and  counterpoise.  Then  submerge 
it  in  water  as  in  the  figure.  The  equilibrium  will 
be  restored  by  placing  25  gm.  in  the  pan  above  the 
Fig.  39  cylinder.  The  cylinder  displaces  25  cm.3  of  water 


Fig.  38 


42  MECHANICS   OF  FLUIDS 

weighing  25  gm.,  and  its  apparent  loss  of  weight  is  25  gm.     The  tem- 
perature of  the  water  should  be  down  near  freezing. 

56.  Equilibrium  of  Floating  Bodies.  —  If  a  body  be  im- 
mersed in  a  fluid,  it  may  displace  a  weight  of  the  fluid 
less  than,  equal  to,  or  greater  than  its  own  weight.     In 
the  first  case,  the  upward  pressure  is  less  than  the  weight 
of  the  body  and  the  body  sinks.     In  the  second  case,  the 

upward  pressure  is  equal  to  the  weight  of  the  body 
and  the  body  is  in  equilibrium.  In  the  third  case, 
the  upward  pressure  exceeds  the  weight  of  the 
body,  and  the  body  rises  until  enough  of  it  is  out 
of  water  so  that  these  forces  become  equal.  In 
liquids  the  buoyancy  is  independent  of  the  depth 
so  long  as  the  body  is  wholly  immersed,  but  it  de- 
creases as  soon  as  the  body  begins  to  emerge  from 
the  liquid.  Hence, 

When  a  body  floats  on  a  liquid  it  sinks  to  such 
a  depth  that  the  weight  of  the  liquid  displaced  equals 
its  own  weight. 

Make  a  wooden  bar  20  cm.  long  and  1  cm.  square  (Fig. 
40).  Drill  a  hole  in  one  end  and  fill  with  enough  shot  to 
give  the  bar  a  vertical  position  when  floating  with  nearly  its 
whole  length  in  water.  Graduate  the  bar  in  millimeters 
along  one  edge,  beginning  at  the  weighted  end,  and  coat  with 
hot  paraffin.  Weigh  the  bar  and  float  it  in  water,  noting  the 
volume  in  cubic  centimeters  immersed.  This  volume  is  equal 
to  the  volume  of  water  displaced ;  and  since  1  cm.8  of  water 

Fig.  40  weighs  1  gm.,  the  weight  of  the  water  displaced  should  equal 
the  volume  of  the  bar  immersed.  This  will  be  found  also 

very  nearly  equal  to  the  weight  of  the  loaded  bar. 

57.  Center  of  Buoyancy.  —  The  center  of  buoyancy  is  the 
center  of  volume  of  the  displaced  liquid.     Two  forces  act  on 
every  floating  body,  its  weight,  which  is  a  force  acting 
downward  with  its  point  of  application  at  the  center  of 


BODIES  IMMERSED  IN  LIQUIDS 


43 


gravity  (§  118)  of  the  body,  and  buoyancy,  a  force  acting 
vertically  upward,  and  with  its  point  of  application  at  the 
center  of  buoyancy.  If 
these  two  forces  are  not  in 
the  same  vertical  line  (Fig. 
41,  D),  the  effect  is  to  ro- 
tate the  floating  body.  If 
the  vertical  line  through 


Fig.  41 


the  center  of  buoyancy  cuts  the  vertical  through  both  cen- 
ters in  the  position  of  equilibrium  A  at  a  point  above  the 
center  of  gravity  G-  (as  in  D),  the  action  is  to  right  the 
body,  and  it  has  angular  stability.  The  object  of  ballast 
in  a  ship  is  to  lower  the  center  of  gravity,  and  to  increase 
its  angular  stability. 

58.    The  Cartesian  Diver.  —  Descartes,  a  French  scientist, 
illustrated  the  principle  of  Archimedes  by  means  of  an 
hydrostatic  toy,  since  called  the  Cartesian  diver.     It  is 
made  of  glass,  is  hollow,  and  has  a  small 
^3HP^     opening   near  the  bottom.     The   figure  is 
partly  filled  with  water  so  that  it  just  floats 
in  a  jar  of  water  (Fig.  42).     When  pressure 
is  applied  to  the  sheet  rubber  tied  over  the 
top  of  the  jar,  it  is  transmitted  to  the  water, 
more  water  enters  the  floating  figure,  and 
the  air  is  compressed.     The  figure  then  dis- 
places less  water  and  sinks.     When  the 
pressure  is  relieved,'  the  air  in  the  diver 
expands  and  forces  water  out  again.     The 
actual  displacement  of  water  is  then  in- 
creased, and  the  figure  rises  to  the  surface.     The  water  in 
the  diver  may  be  so  nicely  adjusted  that  the  little  figure 
will  sink  in  cold  water,  but  will  rise  again  when  the  water 


Fig.  42 


44  MECHANICS   OF  FLUIDS 

has  reached  the  temperature  of  the  room,  and  the  air  in  the 
figure  has  expanded. 

A  good  substitute  for  the  diver  is  a  small  inverted 
homeopathic  vial  in  a  flat  16-oz.  prescription  bottle,  filled 
with  water  and  closed  with  a  rubber  stopper.  By  press- 
ing on  the  sides  of  the  bottle,  it  yields,  the  air  is  com- 
pressed, and  the  vial  sinks. 

A  submarine  boat  is  a  modern  Cartesian  diver  on  a 
large  scale.  It  is  provided  with  tight  compartments, 
into  which  water  may  be  admitted  to  make  it  sink.  It 
may  be  made  to  rise  to  the  surface  by  expelling  some  of 
the  water  by  means  of  strong  pumps. 

III.     DENSITY  AND  SPECIFIC  GRAVITY 

59.  Density.  —  The  density  of  a  substance  is  the  number 
of  units  of  mass  of  it  contained  in  a  unit  of  volume.     In  the 
c.  g.  s.  system  it  is  the  number  of  grams  per  cubic  centi- 
meter.    For  example,  if  4  cm.3  of  a  substance  contain  a 
mass  of  10  gm.,  its  density  is  2.5  gm.  per  cubic  centimeter. 
In  the  English  system  density  is  the  number  of  pounds 
per  cubic  foot,  or  ounces  per  cubic  inch.     By  definition 

7      ._,_         mass 

density  =  — , 

volume 

or  in  symbols, 

d  =  — ;  whence  v  =  -j  and  m  =  vd.     (Equation  2). 

60.  Density  of  Solids.  —  The  density  of  a  solid  body  is 
its  mass  divided  by  its  volume.     Its  mass  may  always  be 
obtained  by  weighing,  but  the  volume  of   an  irregular 
solid  cannot  be  obtained  from  a  measurement  of  its  di- 
mensions.    In  the  c.  g.  s.  system,  however,  the  principle 
of  Archimedes  furnishes  a  simple  method  of  finding  the 


DENSITY  AND   SPECIFIC  GRAVITY 


45 


volume  of  a  solid,  however  irregular  it  may  be ;  for  the 
volume  of  an  immersed  solid  is  numerically  equal  to  its 
loss  of  weight  in  water  (§  55). 


Then 


density 


mass 


loss  of  weight  in  water 


Consider  two  cases  : 


A.  Solids  heavier  than  water.  —  Find  the  mass  of  the  body 
in  air  in  terms  of  grams ;   if  it  is  insoluble  in  water,  find 
its  apparent  loss  of  weight  by  suspending 

it  in  water  (Fig.  43).  This  loss  of  weight 
is  equal  to  the  weight  of  the  volume  of 
water  displaced  by  the  solid  (§  55).  But 
the  volume  of  a  body  in  cubic  centimeters 
is  the  same  as  the  mass  in  grams  of  an  equal 
volume  of  water.  The  mass  divided  by  this 
volume  is  the  density. 

B.  Solids  lighter   than  water.  —  If  the 
body  floats,  its  volume  may  still  be  ob- 
tained by  tying  to  it  a  sinker  heavy  enough  Fl£-  43 

to  force  it  beneath  the  surface.  Let  w1  denote  the  weight 
in  grams  required  to  counterbalance  when  the  body  is  in 
the  air,  and  the  attached  sinker  in  the  water ; 
and  let  wz  denote  the  weight  to  counterbal- 
ance when  both  body  and  sinker  are  under 
water  (Fig.  44)..  Then  obviously  w1  —  w2 
is  equal  to  the  upward  pressure  on  the  body 
alone,  and  is  therefore  numerically  equal  to 
the  volume  of  the  body  •(§  55).  The  mass 
divided  by  this  volume  is  the  density. 

If  the  solid  is  soluble  in  water,  a  liquid  of 
known  density,  in  which  the  body  is  not 
Fig.  44          soluble,  must  be  used  in  place  of  water. 


46  MECHANICS   OF  FLUIDS 

The  buoyancy  of  this  liquid,  divided  by  the  density  of  the 
liquid,  gives  as  before  the  volume  of  the  solid  ;  its  density 
is  then  found  as  before. 

EXAMPLES.  —  First,  for  a  body  heavier  than  water. 

Weight  of  body  in  air 10.5  gm. 

Weight  of  body  in  water      ....       6.3  gm. 
Weight  of  water  displaced    ....      4.2  gm. 

Since  the  density  of  water  is  1  gm.  per  cubic  centimeter,  the  vol- 
ume of  the  water  displaced  is  4.2  cm.8.  This  is  also  the  volume  of 
the  body.  Therefore,  10.5  H-  4.2  =  2.5  gm.  per  cubic  centimeter  is  the 
density. 

Second,  for  a  body  lighter  than  water. 

Weight  of  body  in  air      .     .     .     .     .      4.8  gm. 

Weight  of  sinker  in  water    ....     10.2  gm. 

WTeight  of  body  and  sinker  in  water       8.4  gm. 
The  combined  weight  of  the  body  in  air  and  the  sinker  in  water 
is,  then,  4.8  +  10.2  =  15  gm.     But  when  the  body  is  attached  to  the 
sinker,  their  apparent  combined  weight  is  only  8.4  gm.     Therefore 
the  buoyant  effort  on  the  body  is  15  —  8.4  =  6.6  gm.,  and  this  is  the 
weight  of  the  water  displaced  by  the  body,  and  hence  its  volume  is 
6.6  cm.3.    The  density  is,  then,  4.8  -r-  6.6  =  0.73  gm.  per  cubic  centimeter. 
Third,  for  a  body  soluble  in  water.     Suppose  it  is  insoluble  in  alcohol, 
the  density  of  which  is  0.8  gm.  per  cubic  centimeter. 

Weight  of  body  in  ah; 4.8  gm. 

Weight  of  body  in  alcohol 3.2  gm. 

Weight  of  alcohol  displaced 1.6  gm. 

The  volume  of  alcohol  displaced  is  1.6  -f-  0.8  =  2 
cm.3.  This  is  also  the  volume  of  the  body.  There- 
fore, the  density  of  the  body  is  4.8  -f-  2  =  2.4  gm. 
per  cubic  centimeter. 

61.    Density    of    Liquids.  —  A.    By   the 

specific  gravity  bottle.     A  specific  gravity 
bottle  (Fig.  45)  is  usually  made  to  hold  a 
Fig.  45  definite  mass  of  distilled  water  at  a'speci- 


DENSITY  AND   SPECIFIC  GRAVITY 


47 


Fig.  46 


fied  temperature,  for  example,  25,  50,  or  100  gm.  Its 
volume  is  therefore  25,  50,  or  100  cm.3.  To  use  the 
bottle,  weigh  it  empty,  and  filled  with  the  liquid,  the  den- 
sity of  which  is  to  be  determined.  The  weight  of  the 
liquid  divided  by  the  capacity  of  the  bottle  in  cubic  centi- 
meters (the  number  of  grams)  is  equal  to  the  density  of 
the  liquid. 

B.  By  the  density  bulb.  The  density  bulb  is  a  small 
glass  globe  loaded  with  shot,  and  having  a  hook  for  sus- 
pension (Fig.  46).  To  use  it,  suspend  from  the 
arm  of  a  balance  with  a  fine  platinum  wire,  and 
weigh  first  in  air  and  then  in  water.  The  ap- 
parent loss  of  weight  is  the  weight  of  the  water 
displaced  by  the  bulb  (§  55).  Then  weigh  it 
again  when  suspended  in  the  liquid.  The  loss 
of  weight  is  this  time  the 
weight  of  a  volume  of  the  liquid  equal 
to  that  of  the  bulb.  Divide  this  loss  of 
weight  by  the  loss  in  water,  and  the  quo- 
tient will  be  the  density  of  the  liquid  in 
grams  per  cubic  centimeter,  if  the 
weights  are  in  grams. 

C.  By  the  hydrometer.  The  common 
hydrometer  is  usually  made  of  glass,  and 
consists  of  a  cylindrical  stem  and  a  bulb 
weighted  with  mercury  or  shot  to  make 
it  sink  to  the  required  level  (Fig.  47). 
The  stem  is  graduated,  or  has  a  scale 
inside,  so  that  readings  can  be  taken  at 
the  surface  of  the  liquid  in  which  the 
hydrometer  floats.  These  readings  give 
the  densities  directly,  or  they  may  be 
Fig.  47  reduced  to  densities  by  means  of  an 


48  MECHANICS  OF  FLUIDS 

accompanying  table.  Hydrometers  sometimes  have  a  ther- 
mometer in  the  stem  to  indicate  the  temperature  of  the 
liquid  at  the  time  of  taking  the  reading.  Specially  grad- 
uated instruments  of  this  class  are  used  to  test  milk,  alco- 
hol, acids,  etc. 

62.  Specific  Gravity.  —  The  specific  gravity  of  a  body  is 
the  ratio  between  its  weight  and  the  weight  of  an  equal  vol- 
ume of  water.  If,  for  example,  a  cubic  inch  of  iron  weighs 
7.8  times  as  much  as  a  cubic  inch  of  water,  its  specific 
gravity  is  7.8.  Also,  the  density  of  iron  in  c.  g.  s.  units  is 
7.8  gm.  per  cubic  centimeter;  for,  since  1  cm.3  of  water 
weighs  1  gm.,  1  cm.3  of  iron  weighs  7.8  gm.  Hence,  what- 
ever system  of  units  is  used  to  determine  specific  gravity, 
the  result  will  be  numerically  equal  to  the  density  in  the 
c.  g.  s.  system,  since  specific  gravity  merely  expresses  how 
many  times  heavier  a  body  is  than  an  equal  volume  of 
water.  If  the  density  is  determined  in  c.  g.  s.  units,  the 
numeral  expressing  the  result  is  always  the  specific 
gravity. 

Questions  and  Problems 

1.  Why  does  an  ocean  steamer  draw  more  water  after  entering 
fresh  water  ? 

2.  If  the  Cartesian  diver  should  sink  in  the  jar,  why  will  the 
addition  of  salt  cause  it  to  rise  ? 

3.  What  is  the  density  of  a  body  weighing  15  gm.  in  air  and  10 
gm.  in  water  ?     What  is  its  specific  gravity  ? 

4.  A  hollow  brass  ball  weighs  1  kgm.     What  must  be  its  volume 
so  that  it  will  just  float  in  water? 

5.  What  is  the  density  of  a  body  weighing  20  gm.  in  air  and  16 
gm.  in  alcohol  whose  density  is  0.8  gm.  per  cubic  centimeter? 

6.  A  bottle  filled  with  water  weighed  60  gm.  and  when  empty  20 
gm.     When  filled  with  olive  oil  it  weighed  56.6  gin.     What  is  the 
density  of  olive  oil  ? 


PRESSURE    OF    THE    ATMOSPHERE  49 

7.  A  density  bulb  weighed  75  grn.  in  air,  45  gm.  in  water,  and  21 
gm.  in  sulphuric  acid.     Calculate  the  density  of  the  sulphuric  acid. 

8.  A  piece  of  wood  weighs  96  gm.  in  air,  172  gm.  in  water  with 
sinker  attached.     The  sinker  alone  in  water  weighs  220  gm.     Find 
the  density  of  the  wood. 

9.  A  piece  of  zinc  weighs  70  gm.  in  air,  and  60  gm.  in  water. 
What  will  it  weigh  in  alcohol  of  density  0.8  gm.  per  cubic  centimeter? 

10.  The  mark  to  which  a  certain  hydrometer  weighing  90  gm. 
sinks  in  alcohol  is  noted.     To  make  it  sink  to  the  same  mark  in  water 
it  must  be  weighted  with  22.5   gm.     What   is  the   density  of  the 
alcohol? 

11.  A  body  floats  half  submerged  in  water.     What  is  its  specific 
gravity?     What  part  of  it  will  be   submerged  in   alcohol,  specific 
gravity  0.8  ? 

12.  If  an  iron  ball  weighs  100.4  Ib.  in  air,  what  will  it  weigh  in 
water  if  its  specific  gravity  is  7.8  ? 

13.  What  is  the  specific  gravity  of  a  wooden  ball  that  floats  two 
thirds  under  water? 

14.  A  ferry  boat  weighs  700  tons.     What  will  be  the  displacement 
of  water  if  it  takes  on  board  a  train  weighing  600  tons  ? 

15.  A  liter  flask  weighing  75  gm.  is  half  filled  with  water  and  half 
with  glycerine.     The  flask  and  liquids  weigh  1205  gm.     What  is  the 
density  of  the  glycerine  ?    What  is  its  specific  gravity  ? 

IV.     PRESSURE  OF  THE  ATMOSPHERE 

63.  Weight  of  Air.  —  It  is  only  a  little  more  than  250 
years  since  it  became  definitely  known  that  air  has  any 
weight  at  all.  Even  now  we  scarcely  appreciate  its 
weight. 

Place  a  globe  holding  about  a  liter  (Fig.  48)  on  the  pan  of  a  bal- 
ance and  counterpoise ;  the  stopcock  should  be  open.  Remove  the 
globe  and  force  in  more  air  with  a  bicycle  pump,  closing  the  stopcock 
to  retain  the  air  under  the  increased  pressure ;  the  balance  will  show 
that  the  globe  is  heavier  than  before.  Remove  it  again  and  exhaust  the 
air  with  an  air  pump ;  the  balance  will  now  show  that  the  globe  has  lost 
weight.  A  large  incandescent  lamp  bulb  may  be  used  in  place  of  the 


50 


MECHANICS  OF  FLUIDS 


Fig.  48 


globe  by  first  counterbalancing  and  then  admitting  air  by  punctur- 
ing with  the  very  pointed  flame  of  a  blowpipe.     Thus  air,  though 
invisible,  may  be  put  into  a  vessel  or  removed  like  any 
other  fluid;  and,  like  any  other  fluid,  it  has  weight. 

The  weight  of  a  body  of  air  is  surprisingly 
large.  A  cubic  yard  of  air  at  atmospheric 
pressure  weighs  more  than  2  Ib.  The  air 
in  a  hall  40  ft.  long,  30  ft.  wide,  and  22.5 
ft.  high  weighs  more  than  a  ton.  Precise 
measurements  have  shown  that  air  at  the  tem- 
perature of  freezing  and  under  a  pressure  equal 
to  that  of  a  column  of  mercury  76  cm.  high 
weighs  1.293  gm.  per  liter,  or  0.001293  gm.  per  cubic  centi- 
meter. Hydrogen  under  the  same  conditions  weighs  only 
0.0000895  gm.  per  cubic  centimeter. 

64.  Pressure  produced  by  the  Air.  —  Since  the  air  sur- 
rounding the  earth  has  weight,  it  must  produce  pressure 
on  any  surface  equal  to  the  weight  of  a  column  of  air 
above  it,  just  as  in  the  case  of  a  liquid.  Many  experi- 
ments prove  this  to  be  true. 

Stretch  a  piece  of  sheet  rubber,  and  tie  tightly  over  the  mouth  of  a 
glass  vessel,  as  shown 
in  Fig.  49.  If  the  air 
is  gradually  exhaust- 
ed from  the  vessel, 
the  rubber  will  be 
forced  down  more 
and  more  by  the  pres- 
sure of  the  air  above 
it,  until  it  finally  49 

bursts.     The  depres- 
sion will  be  the  same  in  whatever  direction  the 
rig.  oU  rubber  membrane  may  be  turned. 

Fill  a  common  tumbler  full  of  water,  cover  with  a  sheet  of  paper 
so  as  to  exclude  the  air,  and  holding  the  hand  against  the  paper,  in- 


PRESSURE  OF  THE  ATMOSPHERE  51 

vert  the  tumbler  (Fig.  50).  When  the  hand  is  removed,  the  paper  is 
held  against  the  mouth  of  the  glass  with  sufficient  force  to  keep  the 
water  from  running  out. 

Cut  a  piece  of  glass  tubing  about  20  cm.  long,  and  3  or  4  mm.  bore. 
Dip  it  vertically  into  a  vessel  of  water,  and  close  the  upper  end  with 
the  finger.  The  tube  may  now  be  lifted  out,  and  the 
water  will  remain  in  it.  Figure  51  illustrates  a  pipette ; 
it  is  useful  for  conveying  a  small  quantity  of  liquid  from 
one  vessel  to  another. 

Take  a  quart  tin  can,  which  should  be  closed  except 
a  small  opening,  and  fill  it  about  one  third  full  of  water; 
boil  to  expel  all  the  air,  and  then  close  the  opening  air- 
tight by  solder,  or  in  some  other  equally  effective  way. 
Cool  with  water  to  condense  the  steam  inside.  This 
leaves  a  partial  vacuum,  and  the  pressure  of  the  atmos- 
phere will  cause  the  can  to  collapse. 

65.  The  Rise  of  Liquids  in  Exhausted  Tubes.  —      FiS-  51 
Near  the  close  of  Galileo's  life  his  patron,  the  Duke  of  Tus- 
cany, 'dug  a  deep  well  near  Florence,  and  was  surprised  to 
find  that  he  could  get  no  pump  in  which  water  would  rise 
more  than  about  32  feet  above  the  level  in  the  well.     He 
appealed  to  Galileo  for  an  explanation  ;  but  Galileo  appears 
to  have  been  equally  surprised,  for  up  to  that  time  every- 
body supposed  that  water  rose  in  tubes  exhausted  by  suc- 
tion because  "nature  abhors  a  vacuum."     Pumps  were 
well  known  at  that  time,  and  doubtless  the  Italians  were 
accustomed  to  take  their  lemonade  by  sucking  it  through 
a  straw,  but  no  explanation  of  the  rise  of  liquids  in  ex- 
hausted tubes  had  been  given.     Galileo  suggested  experi- 
ments to  find  out  to  what  limit  nature  abhors  a  vacuum, 
but  he  was  too  old  to  perform  them  himself  and  died  in 
1642,  before  the  problem  was  solved  by  others. 

66.  Torricelli's   Experiment.  —  Torricelli,    a   friend   and 
pupil  of    Galileo,  hit   upon    the   idea   of   measuring   the 
resistance  nature  offers  to  a  vacuum  by  a  column  of  mer- 


52 


MECHANICS   OF  FLUIDS 


cury  in  a  glass  tube  instead  of  a  column  of  water  in  the 
Duke  of  Tuscany's  pump.  The  experiment  was  performed 
in  1643  by  Viviani  under  Torricelli's  direction. 

A  stout  glass  tube  about  a  meter  long,  sealed  at  one  end 
and  filled  with  mercury,  is  stopped  at  the  open  end  with 
the  finger,  and  inverted  in  a  vessel  of  mercury 
in  a  vertical  position  (Fig.  52).  When  the 
finger  is  removed,  the  column  falls  to  a  height 
of  about  76  cm.  The  space  above  the  mercury 
is  known  as  a  Torricellian  vacuum.  The  col- 
umn of  mercury  in  the  tube  is  counterbalanced 
by  the  pressure  of 
the  atmosphere  on 
the  mercury  in  the 
larger  vessel  at  the 
bottom. 

67.  Pascal's  Ex- 
periments. —  To 
Pascal  is  due  the 
credit  of  complet- 
ing the  demonstra- 
tion that  the  weight  of  the 
column  of  mercury  in  the  Tor- 
ricellian experiment  measures 
the  pressure  of  the  atmos- 
phere. He  reasoned  that  if 
the  mercury  is  held  up  simply 
by  the  pressure  of  the  air,  the 
column  should  be  shorter  at 
higher  altitudes  because  there 
is  then  less  air  above  it.  Put  to  the  test  by  carrying  the 
apparatus  to  the  top  of  the  "  Tour  St.  Jacques  "  (Fig.  53), 


Fig.  52 


Fig.  53 


PRESSURE  OF  THE  ATMOSPHERE        53 

at  that  time  the-bell  tower  of  a  church  in  Paris,  his  theory 
was  confirmed.  Desiring  to  carry  the  test  still  further, 
he  wrote  to  his  brother-in-law  to  try  the  experiment  on 
the  Puy  de  Dome,  a  mountain  nearly  1000  m.  high  in 
southern  France.  *  The  result  was  that  the  column  of  mer- 
cury was  found  to  be  nearly  8  cm.  shorter  than  in  Paris. 

Pascal  repeated  the  experiment  with  red  wine  instead  of 
mercury,  and  with  glass  tubes  forty-six  feet  long ;  and  he 
found  that  the  lighter  the  fluid,  the  higher  the  column 
sustained  by  the  pressure  of  the  air.  Further,  a  balloon, 
half  filled  with  air,  appeared  fully  inflated  when  carried  up 
a  high  mountain,  and  collapsed  again  gradually  during  the 
descent.  Thus  the  question  of  the  Duke  of  Tuscany  was 
fully  answered. 

68.  Pressure  of  One  Atmosphere. — The  height  of  the 
column  of  mercury  supported  by  atmospheric  pressure 
varies  from  hour  to  hour ;  it  is  dependent  also  on  the 
altitude  above  the  sea.  Its  height  is  independent  of  the 
cross  section  of  the  tube,  but  to  find  the  pressure  per  unit 
area,  a  tube  of  unit  cross  section  must  be  assumed.  Sup- 
pose an  internal  cross  sectional  area  of  1  cm.2.  The  stand- 
ard height  chosen  is  76  cm.  of  mercury  at  the  temperature 
of  melting  ice  (0°  C.),  and  at  sea  level  in  latitude  45°. 
The  density  of  mercury  at  this  temperature  is  13.596. 
Hence,  standard  atmospheric  pressure,  which  is  the  weight 
of  this  column  of  mercury,  is 

76x13.596  =  1033.3   gm.    per   square  centimeter,  or 
roughly  1  kgm.  per  square  centimeter,  equiva- 
lent to  14.7  Ib.  per  square  inch. 

The  height  of  a  column  of  water  to  produce  a  pres- 
sure of  one  atmosphere  is  76x13.596  =  1033.3  cm.  = 
33.57  ft. 


54  MECHANICS   OF  FLUIDS 

69.  The  Barometer.  —  The  barometer  is  an  instrument 
based  on  Torricelli's  experiment,  and  designed  to  measure 
the  varying  pressure  of  the  atmosphere.  In  its 
simplest  form  it  consists  of  ft  J -shaped  glass 
tube  about  86  cm.  (34  in*)  high,  and  attached 
to  a  supporting  board  (Fig.  54).  The  short 
arm  has.  a  pinhole  near  the  top  for  the  admission 
of  air.  A  scale  is  fastened  by  the  side  of  the 
tube,  and  the  difference  of  readings  at  the  top 
of  the  mercury  in  the  long  and  the  short  arm 
gives  the  height  of  the  mercury  column  sus- 
tained by  atmospheric  pressure.  This  varies 
from  about  73  to  76.5  cm.  for  places  near  sea 
level.  When  accuracy  is  required,  the  baro- 
meter reading  must  be  corrected  for  tempera- 
ture. A  good  barometer  must  contain  pure 
mercury,  and  the  mercury  must  be  boiled  in  the 
glass  tube  to  expel  air  and  moisture. 

70.  The  Aneroid  Barometer.  —  The  aneroid 
barometer  contains  no  liquid.  It  consists  es- 
sentially of  a  shallow  cylindrical  box  B  (Fig. 
55),  from  which  the  air  is  partially  exhausted. 
It  has  a  thin  cover  corrugated  in  circular  ridges 
to  give  it  greater  flex- 
ibility. The  cover  is 
prevented  from  col- 
lapsing under  atmos- 
Fig.  54  r  .  & 

pheric  pressure  by  a 

stiff  spring  attached  to  the  cen- 
ter of  the  cover  at  M.  This 
flexible  cover  rises  and  falls  as 
the  pressure  of  the  atmosphere  Fig.  55 


PRESSURE  OF  THE  ATMOSPHERE  55 

varies,  and  its  motion  is  transmitted  to  the  pointer  by 
means  of  delicate  levers  and  a  chain.  A  scale  graduated 
by  comparison  with  a  mercurial  barometer  is  fixed  under 
the  pointer.  These  instruments  are  so  sensitive  that  they 
readily  indicate  the  change  of  pressure  when  carried  from 
one  floor  of  a  building  to  the  next,  or  even  when  moved  no 
farther  than  from  a  table  to  the  floor. 

71.  Utility  of  the  Barometer.  —  The  barometer  is  a  faithful 
indicator  of  all  changes  in  the  pressure  of  the  atmosphere. 
These  may  be  due  to  fluctuations  in  the  atmosphere  itself, 
or  to  changes  in  the  elevation  of  the  observer. 

The  barometer  is  constantly  used  by  the  Weather  Bu- 
reau in  forecasting  changes  in  the  weather.  Experience 
has  shown  that  barometric  readings  indicate  weather 
changes  as  follows : 

I.  A  rising    barometer   indicates    the    approach   of  fair 
weather. 

II.  A.  sudden  fall  of  the  barometer  precedes  a  storm. 

III.  An  unchanging  high  barometer  indicates  settled  fair 
weather. 

The  difference  in  the  altitude  of  two  stations  may  be 
computed  from  barometer  readings  taken  at  the  two 
places  simultaneously.  Various  complex  rules  have  been 
proposed  to  express  the  relation  between  the  difference  in 
barometer  readings  and  the  difference  in  altitude ;  a  sim- 
ple rule  for  small  elevations  is  to  allow  0.1  in.  for  every 
90  ft.  of  ascent. 

72.  Cyclonic  Storms.  —  Weather  maps  are  drawn   from 
observations  made  at  many  places  at  the  same  time  and 
telegraphed   to   central   stations.     In    this  way  cyclonic 
storms  are  discovered  and  followed.     At  the  center  of  the 


56 


MECHANICS  OF  FLUIDS 


storm  is  the  lowest  reading  of  the  barometer.     Curves  of 
equal   pressure   (called   isobars)    are   traced  around  this 

center  (Fig.  56).  The 
wind  blows  from  areas  of 
higher  pressure  toward 
those  of  lower,  but  in  the 
northern  hemisphere  the 
inflowing  winds  are  de- 
flected toward  the  right  on 
account  of  the  rotation  of 
the  earth.  This  gives  to 
the  storm  a  counter-clock- 
wise rotation,  as  indicated 
by  the  arrows  in  a  weather 
map. 

Fig  56  Cyclonic  storms  usually 

cross  the  northwest  boun- 
dary of  the  United  States  from  British  Columbia,  travel  in 
a  southeasterly  direction  until  they  cross  the  Rocky  Moun- 
tain range,  and  then  turn  northeasterly  toward  the  At- 
lantic coast.  Storms  coming  from  the  Gulf  of  Mexico 
usually  travel  along  the  Atlantic  coast  toward  the  north- 
east. 


Questions  and  Problems 

1.  Why  do  the  ears  sometimes  hurt  when  coming  down  a  fast 
elevator  from  the  top  of  a  tall  building  ? 

2.  What  would  be  the  effect  of  getting  a  little  air  in  the  top  of  a 
barometer  tube  ? 

3.  What  would  happen  if  a  minute  hole  were  made  in  the  top  of 
a  barometer  tube  ? 

4.  Why  is  the  reading  incorrect  if  the  barometer  tube  is  held  in  a 
position  inclined  to  the  vertical  ? 


COMPRESSION  AND  EXPANSION  OF  GASES         57 

5.  A  tube  1  ft.  long  is  closed  at  one  end  by  a  sheet  of  thin  rubber 
tied  over  it  air-tight.     It  is  then  filled  with  mercury  and  inverted  in 
a  vessel  of  mercury  as  in  Torricelli's  experiment.     Why  does   the 
rubber  membrane  settle  down  into  the  tube? 

6.  A  glass  tube  1  ft.  long  is  closed  at  one  end,  filled  with  mercury 
as  in  Torricelli's  experiment,  but  instead  of  resting  on  the  bottom  of 
the  vessel,  it  is  suspended  from  one  arm  of  a  balance.     Does  it  weigh 
more  than  before  it  was  filled?     Give  reason. 

7.  The  barometer  reading  is  75.2  cm.     Calculate  the  atmospheric 
pressure  per  square  centimeter. 

8.  The  barometer  reading  is  29  in.     Calculate  the  atmospheric 
pressure  per  square  inch. 

9.  Calculate  the  buoyancy  of  the  air  for  a  ball  10  cm.  in  diameter 
if  a  liter  of  air  weighs  1.29  gm. 

10.  The  density  of  glycerine  is  1.26  gm.  per  cubic  centimeter.     If 
a  barometer  were  constructed  for  glycerine,  what  would  be  its  reading 
when  the  mercurial  barometer  reads  75  cm.  ? 

11.  When  the  density  of  the  air  is  0.0013  gm.  per  cubic  centimeter, 
how  much  less  will  200  cm.8  of  cork  weigh  in  air  than  in  a  vacuum? 

12.  If  a  barometer  at  the  foot  of  a  tower  reads  29.5  in., 
while  one  at  the  top  reads  29.2  in.,  what  is  the  height  of  the 
tower  ? 

13.  A  bottle  is  fitted  air-tight  with  a  rubber  stopper  and 
a  tube  as  in  Fig.  57.     If  water  be  sucked  out  by  the  tube, 

what  will  happen  when  the  tube  is  re- 
leased? If  air  is  blown  in  through  the 
tube,  what  will  happen  when  the  tube 
is  released  ?  "g.  " 

14.   Fig.  58  represents  a  pneumatic  inkstand, 
Fig.  58  nearly  full  of  ink.     Why  does  the  ink  not  run  out? 

V.     COMPRESSION  AND  EXPANSION  OF  GASES 

73.  Compressibility  of  Air.  —  All  gases  are  compressed 
with  ease.  The  inflation  of  a  toy  balloon,  a  pneumatic 
tire,  or  an  air  cushion  demonstrates  the  easy  compressi- 
bility of  the  air.  It  may  be  shown  in  a  simple  way  by 


58  MECHANICS   OF  FLUIDS 

pushing  a  long  test  tube  under  water  with  its  open  end 
down.  The  deeper  the  tube  is  sunk,  the  higher  the  water 
rises  in  it  and  the  smaller  becomes  the  volume  of  the  in- 
closed air;  also  the  reaction  tending  to  lift  the  tube 
increases. 

The  expansibility  of  air,  or  its  tendency  to  increase  in 
volume  whenever  the  pressure  is  reduced,  is  shown  by  its 
escape  from  any  vessel  under  pressure,  such  as  the  rush  "of 
compressed  air  from  a  popgun,  an  air  gun,  or  a  punctured 
pneumatic  tire.  The  air  in  a  building  shows  the  same 
tendency  to  expand.  When  the  pressure  outside  is  sud- 
denly reduced,  as  in  the  passage  of  a  wave  due  to  an  ex- 
plosion, the  force  of  expansion  of  the  air  within  often 
bursts  the  windows  outward. 

Blow  air  into  the  bottle  (Fig.  57)  through  the  open  tube.  The.  air 
forced  in  bubbles  up  through  the  water  and  is  compressed  within. 
As  soon  as  the  tube  is  released  and  the  pressure  in  it  falls  to  that 
of  the  atmosphere,  the  expansive  force  of  the  imprisoned  air  forces 
water  out  through  the  tube  with  great  velocity. 

The  compression  and  the  expansion  of  air  are  both  illustrated  by 
the  common  pneumatic  door  check  for  light  doors;  also  by  the  air 
dome  on  a  force  pump,  and  the  air  cushion  on  a  water  pipe,  whicfo  is 
usually  carried  a  few  inches  higher  than  the  faucet  so  that  the  air 
confined  in  the  closed  end  may  act  as  a  cushion  to  take  up  any  sudden 
shock  due  to  the  inertia  of  the  water  when  the  stream  is  suddenly 
checked.  The  "pounding"  of  the  pipes  when  the  water  is  suddenly 
turned  off  is  owing  to  the  absence  of  this  air  cushion. 

74.  Boyle's  Law.  —  The  relation  between  the  volume  of 
a  confined  mass  of  air  and  the  pressure  it  sustains  was 
discovered  by  Robert  Boyle  in  Oxford,  and  announced  by 
him  in  1662. 

Boyle's  experiments  were  made  with  a  J-tube  (Fig.  59),  and  they 
extended  only  from  ^  of  an  atmosphere  to  4  atmospheres  pressure. 
The  short  leg  A  was  closed  at  the  top  and  mercury  was  poured  in 
until  it  stood  at  the  same  level  in  both  legs  of  the  tube.  The  air  in 


COMPRESSION  AND  EXPANSION  OF  GASES         59 


the  short  leg  was  then  under  the  same  pressure  as  the  atmosphere 
outside.  Its  volume  was  noted  by  means  of  the  attached  scale,  and 
more  mercury  was  then  poured  into  the  tube.  The  dif- 
ference in  the  level  of  the  mercury  in  the  two  legs  of  the 
tube  gave  the  excess  of  pressure  on  the  inclosed  air  above 
that  of  the  atmosphere.  When  this  difference  amounted 
to  76  cm.,  the  pressure  on  the  gas  in  the  short  tube  was 
2  atmospheres,  and  its  volume  was  reduced  to  one  half. 
When  the  difference  became  twice  76  cm.,  the  pressure 
on  the  enclosed  air  was  3  atmospheres  and  its  volume  be- 
came one  third;  and  so  on. 

This  is  the  law  of  the  compressibility  of 
gases ;  it  is  known  as  Boyle's  law  and  may 
be  expressed  as  follows : 

At  a  constant  temperature  the  volume  of  a 
given  mass  of  gas  varies  inversely  as  the  pres- 
sure sustained  by  it. 

If  the  volume  of  gas  v  under  a  pressure  p 
becomes  volume  v'  when  the  pressure  is  changed  to  p', 
then  by  the  law 

^-  =  -2-;  whence  pv  =pfvr.       (Equation  3) 
v'     p 

In  other  words,  the  product  of  the  volume  of  the  gas  and 
the  corresponding  pressure  remains  constant  for  the  same 
temperature. 

75.  The  Law  Approximate.  —  Extended  investigation  has 
shown  that  Boyle's  law  is  not  rigorously  exact,  even  for 
air  at  moderate  pressures.  In  general,  gases  are  more 
compressible  than  the  law  requires.  Such  gases  as  oxy- 
gen and  nitrogen  show  a  minimum  value  for  the  product 
pv;  beyond  this  minimum  value  an  increase  of  pressure 
causes  the  product  pv  to  increase.  For  hydrogen  the 


Fig.  59 


60 


MECHANICS   OF  FLUIDS 


value  of  pv  is  always  higher  than  it  would  be  if  Boyle's 
law  were  precisely  true.  But  within  moderate  limits  of 
pressure  and  at  ordinary  temperatures,  Boyle's  law  is  ex- 
tremely useful  as  a  working  relation. 

An  example  will  illustrate  its  use :  If  a  mass  of  gas  under  a  pres- 
sure of  72  cm.3  of  mercury  has  a  volume  of  1900  cm.8,  what  would  its 
volume  be  if  the  pressure  were  76  cm.3?  By 
equation  (3),  pv  —  p'v' ; 
hence,  72  x  1900  =  76  x 
v'.  From  this  equation  v' 
=  1800  cmA 


76.    The  Air  Com- 
pressor.  —  A  pump  de- 
signed to  compress  air 
Fig.  60  or  other  gases  under  a 

pressure  of  several  atmospheres  is  shown 
in  section  in  Fig.  60,  and  complete  in  Fig. 
61.      The  piston  is   solid,  and 
ft         there  are  two  metal  valves  at 
the  bottom.     Air  or  other  gas 
is  admitted   through   the   left- 
hand  tube  when  the  piston  rises ; 
when  it  descends,  it  compresses  the  inclosed  air, 
the  pressure  closes  the  left-hand  valve,  and  opens 
the  outlet  valve  on  the  right,  and  the  compressed 
air  is  discharged  into  the  compression  tank. 

A  bicycle  pump  (Fig.  62)  is  an  air  compressor 
of  a  very  simple  type.  The  piston  has  a  cup- 
shaped  leather  collar  <?,  which  permits  the  air  to 
pass  by  into  the  cylinder  when  the  piston  is 
withdrawn,  but  closes  when  the  piston  is  forced 
in.  The  collar  thus  serves  as  a  valve,  allowing 
Fig.  62  the  air  to  flow  one  way  but  not  the  other.  The 


COMPRESSION  AND  EXPANSION  OF  GASES 


61 


compressed  air  is  forced  through  the  tube  forming  the 
piston  rod,  and  the  check  valve  in  the  tire  inlet  prevents 
its  return. 

77.  The  Air  Pump.  —  The  air  pump  for  removing  air  or 
any  gas  from  a  closed  vessel  depends  for  its  action  on  the 
expansive  or  elastic  force  of  the  gas.  The  first  air  pump 
was  invented  by  Otto  von  Guericke,  burgomaster  of 
Vlagdeburg,  about  1650.  In  the  very  simplest  form  the 
valves,  corresponding  with  those  of  the  air  compressor, 
ire  worked  by  the  pressure  of  the  air.  But  though  they 
lay  be  made  of  oiled  silk  and  very  light,  the  pressure  in 
the  vessel  to  be  exhausted  soon  reaches  a  lower  limit  below 
which  it  is  too  small  to  open  the  valve  between  it  and  the 
cylinder  of  the  pump.  On  this  account  automatic  valves, 
operated  mechanically,  are  in  use  on 
the  better  class  of  pumps. 

Figure  63  shows  the  inside  of  one 
of  the  simpler  forms  with  automatic 
valves.  A  piston  P,  with  a  valve 
iS  in  it,  works  in  a  cylindrical  barrel, 
communicating  with  the  outer  air 
by  a  valve  F'at  its  upper  end,  and 
with  the  receiver  to  be  exhausted 
by  the  horizontal  tube  at  the  bot- 
tom. The  valve  Sf  is  carried  by  a 
rod  passing  through  the  piston,  and 
fitting  tightly  enough  to  be  lifted 
when  the  upstroke  begins.  The 
ascent  of  the  rod  is  almost  imme- 
diately arrested  by  a  stop  near  its 


Fig.  63 


upper  end,  and  the  piston  then  slides  on  the  rod  during 
the  remainder  of  the  upstroke.      The  open  valve  S*  allows 


62 


MECHANICS   OF  FLUIDS 


the  air  to  flow  from  E  into  the  space  below  the  piston. 
At  the  end  of  the  upstroke  the  valve  S'  is  closed  by  the 
lever  shown  in  dotted  lines.  During  the  downward  move- 
ment the  valve  S  is  open,  and  the  inclosed  air  passes 
through  it  into  the  upper  part  of  the  cylinder.  The  ascent 
of  the  piston  again  closes  S;  and  as  soon  as  the  air  is  suf- 
ficiently compressed,  it  opens  the  valve  V  and  escapes. 
Each  complete  double  stroke  removes  a  cylinder  full  of 
air ;  but  as  it  becomes  rarer  with  each  stroke,  the  mass  re- 
moved each  time  grows  less. 

78.    Experiments  with  the  Air  Pump.  —  1.  Expansibility  of 

air. 

(a)  Football.  Fill  a  small  rubber  football  half  full  of  air,  and 
place  under  a  bell  jar  on  the  table  of  the  air  pump.  When 
the  air  is  exhausted  from  the  jar,  the  football  expands 
until  it  is  free  from  wrinkles.  A  toy  balloon  may  be  sub- 
stituted. 

(b)  Bolthead.  A  glass  tube  with  a  large  bulb  blown  on 
one  end  (Fig.  64)  is  known  as  a  bolthead.  The  stem 
passes  air-tight  through  the  cap  of  the  bell  jar,  and  dips 
below  the  surface  of  the  water  in  the  inner  vessel.  When 
the  air  is  exhausted  from  the  jar,  the  air  in  the  bolthead 
expands  and  escapes  in  bubbles 
through  the  water.  Readmission 
of  air  into  the  jar  restores  the 
Fig.  64  pressure,  and  drives  water  into 

the  bolthead. 

2.   Air  pressure,     (a)   Downward.     Wet   a 

piece  of  parchment,  and  tie  it  tightly  over  the 

mouth  of  a  glass  cylinder  (Fig.  65).     A  piece 

of  parchment  paper  may  be  pasted  over  the 

cylinder  instead.     When  the  air  is  exhausted, 

the  parchment  or  the  paper  will  break  with  a 

loud  report. 


Fig.  65 


(6)   The  vacuum  fountain.     A  tall  glass  vessel  has  an  inner  jet  tube 
which  may  be  closed  on  the  outside  with  a  stopcock.     (A  stout  bottle 


COMPRESSION  AND  EXPANSION  OF  GASES 


63 


with  a  rubber  stopper  and  a  jet  tube  may  be  substituted.)  Exhaust 
the  air,  place  the  opening  into  the  jet  tube  in  water,  and  open  the 
stopcock.  The  water  is  forced  by  atmospheric  pres- 
sure into  the  exhausted  tube  like  a  fountain  (Fig. 


(c)  Upward  pressure.  A  strong  glass  cylinder  is 
fitted  with  a  piston,  and  is  supported  on  a  tripod 
(Fig.  67).  The  brass  cover  of  the  cylinder  is  con- 
nected with  the  air  pump  by  a  thick  rubber  tube. 
When  the  air  is  exhausted,  the 
piston  is  lifted  by  atmospheric 
pressure,  and  carries  the  heavy 
attached  weight. 

(d)  The  Magdeburg  hemi- 
spheres. This  historical 
piece  of  apparatus  was 
designed  by  Otto  von 
Guericke  to  exhibit 
the  great  pressure  of 
the  atmosphere  (Fig. 
The  lips  of  the 


Fig.  67 


68). 


Fig.  66 


two  parts  are  accurately  ground  to  make  an 
air-tight  joint  when  greased.  When  they  are  brought  together  and 
the  air  is  exhausted,  it  requires  considerable  force  to  pull 
them  apart.  The  original  hemispheres  of  von  Guericke 
were  about  1 .2  ft.  in  diameter,  and  the  atmospheric  pres- 
sure holding  them  together  was 
about  2400  Ib. 


Fig.  69 


79.  Buoyancy  of  the  Air.  — 
A  small  beam  balance  has  at- 
tached to  one  arm  a  hollow 
closed  brass  globe  ;  it  is  coun- 
terbalanced in  air  by  a  solid 
brass  weight  on  the  other  arm. 
(A  large  cork  or  a  glass  float  may  be 
substituted  for  the  globe,  and  a  piece  of 
lead  for  the  brass  weight.)  When  the 


Fig.  68 


64  MECHANICS   OF  FLUIDS 

balance  is  placed  under  a  bell  jar,  and  the  air  is  exhausted, 
the  globe  overbalances  the  solid  weight  (Fig.  69). 

The  apparatus  is  called  a  baroscope.  It  shows  that  the 
atmosphere  exerts  an  upward  pressure  or  buoyant  force 
on  bodies  immersed  in  it ;  for  the  principle  of  Archimedes 
applies  to  gases  as  well  as  to  liquids.  The  buoyancy  of 
the  atmosphere  is  equal  to  the  weight  of  the  air  displaced 
by  a  body.  Whenever  a  body  is  less  dense  than  the 
weights,  it  weighs  more  in  a  vacuum  than  in  the  air. 

80.  Balloons  and  airships  also  illustrate  the  buoyancy 
of  the  air.  A  soap  bubble  and  a  toy  balloon  filled  with 
air  fall  because  they  are  heavier  than  the  air  displaced ; 
but  a  bubble  filled  with  hydrogen  or  coal  gas  rises  in  the 
air.  Its  buoyancy  is  greater  than  its  weight,  including 
the  inclosed  gas.  The  weight  of  a  balloon  with  its  car 
and  contents  must  be  less  than  that  of  the  air  displaced 
by  it.  The  essential  part  of  a  balloon  is  a  silk  bag,  var- 
nished to  make  it  air-tight ;  it  is  filled  either  with  hydro- 
gen or  with  illuminating  gas.  A  cubic  meter  of  hydrogen 
weighs  about  0.09kgm.,  a  cubic  meter  of  illuminating  gas, 
0.75  kgm.,  while  a  cubic  meter  of  air  weighs  1.29  kgm. 
(§  63).  With  hydrogen  the  buoyancy  is  1.29-0.09  = 
1.2  kgm.  per  cubic  meter;  with  illuminating  gas  it  is 
1.29  —  0.75  =  0.54  kgm.  per  cubic  meter.  The  latter  is 
more  commonly  used  because  it  is  much  cheaper. 

A  balloon  is  not  fully  inflated  to  start  with,  but  it 
expands  as  it  rises  because  the  pressure  of  the  air  on  the 
outside  diminishes.  The  buoyancy  then  decreases  only 
slowly  as  the  balloon  ascends  into  a  rarer  atmosphere.  If 
it  were  fully  inflated  at  the  start,  the  inside  pressure  of 
the  gas  as  the  balloon  ascends  would  be  greater  than  the 
diminishing  atmospheric  pressure,  and  the  bag  would 


PROBLEMS  65 

almost    certainly   burst    before    any   great   altitude    was 
reached. 

One  of  the  most  noted  long  distance  balloon  journeys  was  that  of 
Count  de  la  Vaulx,  in  1900,  who  traveled  from  Paris  into  Russia,  a 
distance  of  1193  miles,  in  35  hr.  45  min.  The  greatest  altitude 
reached  was  18,700  ft.  The  balloon  United  States,  which  won  the 
first  international  race  at  Paris  in  1906,  was  filled  with  more  than 
2000  cubic  meters  of  illuminating  gas,  with  a  lifting  force  of  1000 
kgm.,  or  2200  Ib. 

Problems 

1.  A  certain  mass  of  gas  under  a  pressure  of  one  atmosphere  has 
a  volume  of  6000  1.     To  how  many  atmospheres  must  the  pressure  be 
increased  to  reduce  the  volume  to  1000  liters? 

2.  A  steel  tank  having  a  capacity  of  3  cu.  ft.  is  filled  with  oxygen 
under  a  pressure  of  10  atmospheres.     How  much  gas  is  in  the  tank 
estimated  at  standard  atmospheric  pressure  ? 

3.  A  liter  of  air  at  0°  C.  and  under  a  pressure  of  76  cm.  weighs 
1.29  gm.     How  much  would  a  liter  weigh  if  the  barometric  pressure 
were  reduced  to  72  cm.?     (The  mass  varies  directly  as  the  pressure.) 

4.  An  open  vessel  contains  200  gm.  of  air  when  the  pressure  is 
74  cm.     How  much  would  it  contain  if  the  pressure  were  76  cm.  ? 

5.  The  volume  of  hydrogen  collected  over  mercury  in  a  graduated 
cylinder  was  50  cm.8,  the  mercury  standing  15  cm.   higher   in   the 
cylinder  than  outside  of  it.     The  reading  of  the  barometer  was  75 
cm.     How  many  cubic  centimeters  of  hydrogen  would  there  be  at  a 
pressure  of  76  cm.  ? 

SUGGESTION.  The  height  of  the  mercury  in  the  cylinder  above  the  surface 
of  the  mercury  outside  must  be  subtracted  from  the  barometer  reading  to  get 
the  pressure  of  the  gas  in  the  cylinder. 

6.  A  test  tube  is  forced  down  into  water  with  its  open  end  down, 
until  the  air  in  it  is  compressed  into  the  upper   half  of   the  tube. 
How  deep  down  is  the  tube  if  the  barometer  stands  at  30  in.  ?     (The 
specific  gravity  of  mercury  may  be  taken  as  13.6.) 

7.  What  part  of  the  air  is  left  in  a  bell  jar  on  an  air  pump  when 
the  mercury  in  the  gauge  is  2  in.  higher  on  one  side  than  on  the 
other,  if  the  barometer  stands  at  30  in.? 


66 


MECHANICS   OF  FLUIDS 


8.  What  will  be  the  difference  in  the  heights  of  the  mercury 
columns  in  the  gauge  when  7^-0  of  the  air  is  left  in  the  receiver? 

9.  With   what  volume   of  illuminating,  gas   must  a  balloon  be 
filled  in  order  to  rise,  if  the  empty  balloon  and  its  contents  weigh  540 
kgm.  ? 

10.   A  mass  of  iron,  density  7.8,  weighs  2  kgm.  in  air.     How  much 
will  it  weigh  in  a  vacuum  ? 

VI.     PNEUMATIC  APPLIANCES 

81.  The  Siphon.  —  The  siphon  is  a  U-shaped  tube  em- 
ployed to  convey  liquids  from  one  vessel  to  another  at 
a  lower  level  by  means  of  atmospheric 
pressure.  If  the  tube  is  filled  and  is 
placed  in  the  position  shown  in  Fig. 
70,  the  liquid  will  flow  out  of  the 
vessel  and  be  discharged  at  the  lower 
level  D. 

If  the  liquid  flows  outward  past  the 
highest  point  of  the  tube  in  the  direc- 
tion BO,  it  is  because  the  pressure  on 
the  liquid  outward  is  greater  than  the 
pressure  in  the  other  direction.  Now 
the  outward  pressure  at  the  top  is  the 
pressure  of  the  atmosphere  minus  the 
weight  of  the  column  of  liquid  AB  ; 
while  the  pressure  inward  is  atmospheric  pressure  minus 
the  weight  of  the  column  DO.  Hence,  the  pressure  in- 
ward is  less  than  the  pressure  outward  by  the  weight  of 
a  column  of  the  liquid  equal  in  height  to  the  difference 
between  AB  and  DO.  The  siphon  will  cease  to  act  when 
the  liquid  reaches  the  lower  end  of  the  shorter  arm  AB, 
or  if  the  liquid  flows  into  another  vessel,  when  the  level  is 
the  same  in  the  two  vessels.  It  will  also  fail  to  act  in 


L 


Fig.  70 


PNEUMATIC  APPLIANCES 


67 


71 


water  when  the  bend  of  the  tube  is  more  than  about  33 
feet  above  the  surface  of  the  water  at  A. 

An  intermittent  siphon  (Fig.  71)  has  its  short  arm  inside  a  vase 
and  its  long  arm  passing  through  the  bottom.  The  vase  will  hold 
water  until  its  level  reaches  the  top  of  the  bend  of  the 
siphon.  It  then  discharges  and  empties  the  vessel,  if 
it  discharges  faster  than  it  is  filled.  Again  the  water 
rises  in  the  vase,  and  the  siphon 
again  empties  it.  Intermittent 
springs  are  supposed  to  operate 
on  the  same  principle. 

A  so-called  vacuum  siphon 
may  be  made  with  a  Florence 
flask  and  glass  tubing  (Fig.  72). 
The  flask  is  partly  filled  with 
water,  and  the  apparatus  is  then 
inverted  as  shown.  The  water 
enters  the  flask  as  a  jet.  If  a 
piece  of  rubber  tubing  is  attached  to  the  longer 
arm,  the  jet  will  rise  as  the  end  of  the  tubing  is 
lowered.  A  portion  of  the  water  runs  out  at 
first,  producing  a  partial  vacuum  inside. 

A  siphon  made  of  glass  tubing  about  2  mm. 
in  diameter,  may  be  set  up  with  mercury  as  the 
liquid.  If  it  is  set  in  action  under  a  tall  bell  jar  on  the  air  pump,  it 
will  stop  working  when  the  air  is  exhausted  from  the  jar,  but  will 
begin  again  when  the  air  is  admitted. 

The  water  of  an  S-trap,  in  common  use  under  sinks  and  washbowls, 
may  be  siphoned  off  when  the  discharge  pipe  is  filled  with  water  for  a 
short  distance  below  the  trap,  unless  the  trap  is  ventilated  at  the  top 
of  the  S. 

82.  The  Lift  Pump.  —  The  common  suction  pump  acts 
by  the  pressure  of  the  air ;  it  is,  in  fact,  a  simple  form  of 
air  pump;  but  it  was  in  use  2000  years  before  the  air 
pump  was  invented.  The  first  few  strokes  serve  merely 
to  exhaust  air  from  the  pipe  below  the  valve  F(Fig.  73)  ; 
the  pressure  of  the  air  on  the  water  in  the  well  or  cistern 


Fig.   72 


68 


MECHANICS   OF  FLUIDS 


then  forces  it  up  the  pipe  S,  and  finally  through  the  valve 
V.  After  that,  when  the  piston  descends,  the  valve  Fcloses 
under  water,  arid  water  passes 
through  the  valve  V  above  the 
piston.  The  next  upstroke  lifts 
the  water  to  the  level  of  the  spout. 
Since  the  pressure  of  the  air  lifts 
the  water  to  the  highest 
point  to  which  the  pis- 
ton ascends,  it  is  obvi- 
ous that  this  point  can 
not  be  more  than  the 
limit  of  about  33  ft. 
above  the  water  in  the 
well.  Practically  it  is 
less  on  account  of  leak- 
age through  the  imper- 
fect valves.  The  prim- 
ing of  a  pump  by  pouring  in  a  little  water  to 
start  it  serves  to  wet  the  valves  and  make  them 
air-tight. 

For  deep  wells  the  common  pump  is  modified 
by  placing  the  piston  and  the 
valves  v  and  v'  far  down 
the  well;  a  long  pump  rod 
serves  to  lift  the  water  from 
the  piston  to  the  spout  (Fig. 
74). 


73 


sm 

Fig.  74 


83.    The    Force    Pump.  —  The    force 
pump  (Fig.  75)  is  used  to  deliver  water 
under  pressure,  either  at  a  point  higher 
Fig.  75  than  the  pump  into  pipes,  as  in  the  fire 


PNEUMATIC  APPLIANCES 


69 


engine  and  the  hydraulic  press,  or  into  boilers  under  pres- 
sure of  the  steam.  The  construction  is  obvious  from  the 
figure. 

The  air  dome  D  is  added  to  secure  a  continuous  flow 
through  the  delivery  pipe  d.  Water  flows  out  through  v' 
only  while  the  piston  is  descending ;  without  the  air  dome, 
therefore,  water  would  flow  through  the  pipe  d  only  during 
the  downstroke  of  the  piston ;  but  the  water  under  pres- 
sure from  the  piston  enters  the  dome  and  compresses  the 
air.  The  elastic  force  of  the  air  drives  the  water  out  again 
as  soon  as  vr  closes.  Thus  the  flow  is  continuous. 

The  pump  of  a  steam  fire  engine  is  double  acting,  that  is, 
it  forces  water  out  while  the  piston  is  moving  in  either 
direction ;  so  also  are  pumps  for  waterworks  and  mines. 

84.  The  Air  Brake.  —  The  well-known  Westinghouse  air  brake 
is  operated  by  compressed  air.  In  Fig.  76,  P  is  the  train  pipe  leading 
to  a  large  reservoir  at  the 
engine,  in  which  an  air 
compressor  maintains  a 
pressure  of  about  75  Ib. 
per  square  inch.  So  long 
as  this  pressure  is  applied 
through  P,  the  automatic 
valve  V  maintains  com- 
munication between  P 
and  an  auxiliary  reser- 
voir R  under  each  car, 
and  at  the  same  time 
shuts  off  air  from  the 

brake    cylinder    C.     But  „.     ^, 

J  Fig.  76 

as   soon  as  the  pressure 

in  P  falls,  either  by  the  movement  of  a  lever  in  the  engineer's  cab  or 
by  the  accidental  parting  of  the  hose  coupling  k,  the  valve  Fcuts  off 
P  and  connects  the  reservoir  R  with  the  cylinder  C.  The  pressure 
on  the  piston  in  C  drives  it  powerfully  to  the  left  and  sets  the  brake 
shoes  against  the  wheels.  As  soon  as  air  from  the  main  reservoir  is 


70  MECHANICS   OF  FLUIDS 

again  admitted  to  the  pipe  P,  the  valve  V  re-establishes  communica- 
tion between  P  and  R,  and  the  confined  air  in  C  escapes.  The  brakes 
are  released  by  the  action  of  the  spring  S  in  forcing  the  piston  back 
to  the  right. 

85.  Other  Applications  of  the  Air  Pump  and  the  Air  Com- 
pressor.—  The  air  pump  and  the  air  compressor  are  extensively  used 
in  industry.  Sugar  refiners  employ  the  air  pump  to  reduce  the  boiling 
point  of  the  syrup ;  manufacturers  of  soda  water  use  a  compressor  to 
charge  the  water  with  carbon  dioxide ;  in  pneumatic  dispatch  tubes, 
now  extensively  used  for  carrying  small  packages,  both  pumps  are 
used,  one  to  exhaust  the  air  from  the  tube  in  front  of  the  closely 
fitting  carriage,  and  the  other  to  compress  air  in  the  tube  behind  it, 
so  as  to  propel  the  carriage  writh  great  velocity.  The  air  compressor 
is  employed  to  make  a  forced  draft  for  steam  boilers,  to  ventilate 
buildings,  and  to  operate  machinery  in  places  difficult  of  access,  as  in 
mines,  where  it  furnishes  fresh  air  as  well  as  power.  It  is  employed 
also  in  the  pneumatic  caisson  for  making  excavations  and  laying 
foundations  under  water.  The  caisson  is  a  large  heavy  air  chamber 
which  sinks  as  the  soft  earth  is  removed  from  within.  When  its 
bottom  is  below  water  level,  air  is  forced  in  under  sufficient  pressure 

to  prevent  the  entrance  of  water. 

Access  to  it  is  gained  by  air-tight 

locks. 

Compressed  air  is  frequently 

used  for  operating  railway  sig- 
Fig.  77  nals,  and  to  control  automatic 

heating  and  ventilating  appli- 
ances. Pneumatic  tools  are  used  for  calking  seams  and  joints,  for 
stone  cutting,  chipping  iron,  and  riveting.  Figure  77  shows  a  riveting 
hammer ;  A  is  the  air  pipe,  £  the  trigger  for  controlling  the  airland 
C  the  hammer. 


CHAPTER  IV 

MOTION 
I.     MOTION  IN  STRAIGHT  LINES 

86.  Types  of  Motion.  —  Motion  is  the  change  in  the  rela- 
tive position  of  a  body  with  respect  to  some  point,  line, 
or  place  of  reference.     A  body  is  at  rest  when  its  relative 
position  remains  unchanged.      All  rest  and  motion  are 
relative  only,  since  there  are  no  fixed  points  or  lines  in 
space  to  which  absolute  motion  may  be  referred. 

The  moving  about  of  a  person  on  a  ship  is  relative  to  the  vessel; 
the  movement  of  the  ship  across  the  ocean  is  relative  to  the  earth's 
surface ;  the  daily  motion  of  the  earth's  surface  is  relative  to  its  axis 
of  rotation  ;  the  motion  of  the  earth  as  a  whole  is  relative  to  the  sun ; 
while  the  sun  itself  is  drifting  with  other  stars  through  space. 

The  motion  of  a  body  is  rectilinear  when  it  moves  along 
a  straight  line ;  curvilinear  when  its  path  is  a  curved  line. 

Then  there  is  also  simple  harmonic  motion,  exemplified 
by  the  to-and-f  ro  swing  of  a  pendulum  ;  end  rotary  motion 
about  an  axis,  such  as  the  rotation  of  the  earth  on  its  axis, 
and  that  of  the  pulley  and  armature  of  a  stationary  elec- 
tric motor.  The  motion  of  a  carriage  wheel  along  a  level 
road,  and  that  of  a  ball  along  the  floor  of  a  bowling  alley 
combine  motion  of  rotation  with  rectilinear  motion. 

87.  Velocity.  —  Velocity  is  the  rate  of  motion,  that  is,  it 
is  the  distance  traversed  per  unit  of  time.     In  expressing 
a  velocity  the  unit  of  time  must  be  given  as  well  as  the 
number  denoting  the  velocity;  for  example,  44.7  centi- 

71 


72  MOTION 

meters  per  second,  5280  feet  per  minute,  or  60  miles  per 
hour.  These  are  all  different  expressions  for  the  same 
velocity  or  rate  of  motion. 

If  the  motion  is  over  equal  distances  in  equal  and  suc- 
cessive periods  of  time,  the  motion  is  uniform,  and  the 
velocity  is  constant;  otherwise  the  velocity  is  variable,  and 
the  motion  is  either  accelerated  or  retarded.  When  there 
is  a  gain  in  velocity  from  instant  to  instant,  the  motion  is 
said  to  be  accelerated  ;  when  there  is  a  loss,  it  is  retarded. 
A  retardation  is  counted  as  a  negative  acceleration. 
When  an  electric  street  car  starts  from  rest,  its  motion  is 
accelerated  until  it  reaches  top  speed  ;  in  going  up  a 
grade,  or  nearing  a  sharp  curve,  or  slowing  down  to  stop 
at  a  station  or  crossing,  its  motion  is  retarded. 

88.  Uniform  Motion.  —  Since  velocity,  denoted  by  v,  is 
the  distance  traversed  per  unit  of  time,  then  in  uniform 
motion  it  must  be  equal  to  the  whole  distance  s  traversed 
divided  by  the  number  of  units  of  time  t  spent  in  travers- 
ing that  distance.  Whence 

7     .         distance 

velocity  =  — -, • 

time 

In  symbols  this  is  written,  v  =  - ;  from  which 

t 

o 

s  =  vt  and  t=  - .     .     .     (Equation  4) 

These  three  simple  equations  express  in  order  the  fol- 
lowing relations  between  velocity,  distance,  and  time  in 
uniform  mption: 

1.  The  velocity  in  uniform  motion  is  the  quotient  of  the 
whole  distance  traversed  by  the  time  of  traversing  it. 

2.  The    distance    traversed   in    uniform    motion    is    the 
product  of  the  velocity  and  the  time. 


MOTION  IN   STRAIGHT  LINES  73 

3.  The  time  required  to  traverse  a  given  distance  in  uni- 
form motion  is  the  quotient  of  the  distance  by  the  velocity. 

EXAMPLE.  A  railway  train  runs  uniformly  covering  660  ft.  in  10 
min.  Then  v  =  -6T%°-  =  66  ft.  per  minute  or  f  mi.  per  hour.  The 
distance  s  =  66  x  10  =  660  ft.  The  time*  =  W  =  10  min. 

89.  Velocity  at  any  Instant.  —  When  the  motion  is  vari- 
able, the  velocity  of  a  body  at  any  instant  is  the  distance 
it  would  travel  in  the  next  unit  of  time  if  at  that  instant 
its  motion  wereXto  become  uniform. 

For  example  :\  The  velocity  of  a  falling  body  at  any 
moment  is  the  distance  it  would  fall  during  the  following 
second,  if  the  attraction  of  the  earth  and  the  resistance  of  the 
air  were  loth  to  be  withdrawn.  The  velocity  of  a  ball  as  it 
leaves  the  muzzle  of  a  gun  is  the  distance  it  would  pass 
over  in  the  second  following  if  from  that  instant  it  should 
continue  to  move  for  a  second  without  any  change  in  speed. 
Actually  the  motion  of  the  body  and  the  ball  for  the  suc- 
ceeding second  is  variable  ;  the  inquiry  is,  what  would  be 
the  velocity  if  the  motion  were  invariable  ? 

90.  Acceleration.  —  When  a  train  runs  a  mile  a  minute 
for  several  minutes,  it  moves  with  uniform  velocity;  but 
when  it  is  starting  or  slowing  down,  it  is  said  to  be  accel- 
erated.    If  the  velocity  increases,  the  acceleration  is  posi- 
tive ;  if  it  decreases,  it  is  negative.     A  falling  body  goes 
faster  and  faster  ;  it  has  a  positive  acceleration.     A  body 
thrown   upward   goes   more   and   more    slowly;     it    has 
a  negative  acceleration.     A  loaded  sled  starts  from  rest 
at  the  top  of  a  long  hill;  it  gains  in  velocity  as  it  descends 
the  hill ;  it  has  a  positive  acceleration.     When  it  reaches 
the  bottom,  it  loses  velocity  and  is  retarded,   or  has  a 
negative  acceleration,  until  it  stops.     Acceleration  is  the 
rate  of  change  of  velocity.     It  is  the  change  in  velocity  per 
unit  of  time. 


74  MOTION 

If  the  change  in  velocity  is  the  same  from  second  to 
second,  the  motion  is  uniformly  accelerated.  The  best 
example  we  have  of  uniformly  accelerated  motion  is  that 
of  a  falling  body,  such  as  a  stone  or  an  apple.  Neglect- 
ing the  resistance  of  the  air,  its  gain  in  velocity  is  9.8  m. 
per  second  for  every  second  it  falls.  Its .  acceleration  is 
therefore  9.8  m.  per  second  per  second;  in  other  words,  it 
gains  in  velocity  9.8  m.  per  second  for  every  second  of 
time.  This  is  equivalent  to  an  increase  in  velocity  of  588 
m.  per  second  acquired  in  a  minute  of  time.  The  unit  of 
time  enters  twice  into  every  expression  for  acceleration,  the 
first  to  express  the  change  in  velocity,  and  the  second  to 
denote  the  interval  during  which  this  change  takes  place. 

If  an  automobile  starts  from  rest  and  increases  its  speed 
one  foot  a  second  for  a  whole  minute,  its  velocity  at  the 
end  of  the  minute  is  60  ft.  per  second.  Since  it  gains  in 
one  second  a  velocity  of  one  foot  a  second,  and  in  one 
minute  a  velocity  of  60  ft.  a  second,  its  acceleration  may 
be  expressed  either  as  one  foot  per  second  per  second,  or 
as  60  ft.  per  second  per  minute.  Its  velocity  is  con- 
stantly changing;  its  acceleration  is  constant. 

91.  Velocity  in  Uniformly  Accelerated  Motion.  —  Suppose 
a  body  to  move  from  rest  in  any  given  direction  with  a 
constant  acceleration  of  5  ft.  per  second  per  second.  Its 
velocity  at  the  end  of  the  first  second  will  be  5  ft.  per 
second  ;  at  the  end  of  two  seconds,  2  x  5  ft.  ;  at  the  end 
of  three  seconds,  3x5  ft.  ;  and  at  the  end  of  t  seconds, 
t  x  5  ft.  per  second  ;  that  is, 

final  velocity  —  time  x  acceleration, 
or  in  symbols, 

v  =  ta  ;  whence  a  —  -  -     .     .   (Equation  5) 
~ 


MOTION  IN   STRAIGHT  LINES 


75 


92.  Distance  traversed  in  Uniformly  Accelerated  Motion.  — 
If  we  can  find  the  mean  or  average  velocity  for  any  period 
of  t  seconds,  the  distance  s  traversed  in  t  seconds  may  be 
found  precisely  as  in  the  case  of  uniform  motion  (§  88). 
For  a  body  starting  from  rest  with  an  acceleration  of  5 
ft.  per  second  per  second,  for  example,  its  velocity  at  the 
end   of   four   seconds   is   4  x  5   ft.    per   second,  and   the 
average  velocity  for  the  four  seconds  is  one  half  of  4  x  5, 
or  2  x  5  ft.  per  second,  the  velocity  at  the  middle  of  the 
period.     So  at  the  end  of  t  seconds  the  average  velocity  is 
\ta  ft.  per  second.     Then  we  have 

distance  =  average  velocity  x  time, 

or  in  symbols, 

s  =  ±ta  xt=  \a&     .     .     (Equation  6) 

From  the  last  two  articles  we  derive  the  following  laws 
for  uniformly  accelerated  motion : 

I.  In  uniformly  accelerated  motion  the  velocity  at- 
tained in  any  given  time  is  proportional  to  the  time. 

II.  In  uniformly  accelerated  motion  the  distance  trav- 
ersed is  proportional  to  the  square  of  the  time. 

93.  Uniformly  Accelerated  Motion  Illustrated.  —The  oldest 
method  of  demonstrating  uniformly   accelerated   motion 
was    devised    by 

Galileo.  It  con- 
sists of  an  in- 
clined plane  two 
or  three  meters 
long  (Fig.  78), 
made  of  a  straight 

board  with  a  shal-  Fig.  78 

low  groove,  down  which  a  marble  may  roll  slowly  enough 
to  permit  the  distances  to  be  noted.     For  measuring  time, 


76 


MOTION 


a  clock  beating  seconds,  or  a  metronome,  may  be  used. 
Assume  a  metronome  as  shown  in  the  figure.  One  end 
of  the  board  should  be  elevated  until  the  marble  will 
roll  from  a  point  near  the  top  to  the  bottom  in  three 
seconds. 

Hold  the  marble  in  the  groove  against  a  straightedge 
in  such  a  way  that  it  may  be  quickly  released  at  a  click  of 
the  metronome.  Find  the  exact  position  of  the  straight- 
edge near  the  top  of  the  plane  from  which  the  marble  will 
roll  to  the  bottom  and  strike  the  block  there  so  that  the 
blow  will  coincide  with  the  third  click  of  the  metronome 
after  the  release  of  the  marble.  Measure  exactly  the  dis- 
tance between  the  upper  edge  of  the  straightedge  and  the 
block  at  the  bottom  and  call  it  9d.  Next,  since  distances 
are  proportional  to  the  square  of  the  times,  let  the  straight- 
edge be  placed  at  a  distance  of  4  d  from  the  block ;  the 
marble  released  at  this  point  should  reach  the  block  at  the 
second  click  of  the  metronome  after  it  starts.  Finally, 
start  the  marble  against  the  straightedge  at  a  distance  d 
from  the  block  ;  the  interval  this  time  should  be  that  of 
one  beat  of  the  metronome,  which  should  be  adjusted  to 
beat  approximately  seconds. 

TABULAR  EXHIBIT 


NUMBER  OF 
SECONDS,  t 

WHOLE  DISTANCE 
FALLEN,  8 

DISTANCE  IN 
SUCCESSIVE  SECONDS 

VELOCITIES 
ATTAINED,  v 

1 

d 

d 

2d 

2 

±d 

3d 

±d 

3 

9d 

5d 

6d 

4 

IQd 

7d 

Sd 

The  third  column  is  derived  by  subtracting  the  suc- 
cessive numbers  of  the  second.     To  get  the  fourth  column, 


PROBLEMS 


77 


v=2d 


>16d 


Id 


we  notice  that  if  t  is  one  second  in  equation  (6),  then 

«  =  1  a  ;  that  is,  the  distance  traversed  in  the  first   second 

is  one  half  the  acceleration.     But 

the    acceleration     is    the     same 

as     the     velocity    acquired    the 

first  second.     Hence  s  =  J  v  and 

d—\v.     Therefore  the  velocity  >9d 

at  the  end  of  the  first  second  on 

the  inclined  plane  is  2d.     Since  35d 

by    equation    (5)    the    velocities 

are  proportional  to  the  time,  the 

succeeding  velocities  are  4d,  6d, 

etc. 

The  numbers  in  the  second 
column  show  that  the  distances 
traversed  are  proportional  to  the 
squares  of  the  time  [compare 
equation  (6)];  those  of  column 
three  show  that  the  distances  FiS- 79 

in  successive  seconds  are  as  the  odd  numbers  1,  3,  5,  etc. 
The  results  may  be  shown  graphically  as  in  Fig.  79. 

Problems 

NOTE.     For  the  relation  between  the  circumference  of  a  circle  and  its 
diameter,  see  the  Mensuration  Table  in  the  Appendix. 

1.  An  aviator  drives  his   aeroplane  through  the  air  a  distance  of 
500  km.  in  8  hr.  20.  min.     What  was  his  average  speed  per  minute? 

2.  The  engine  drives  a  boat  downstream  at  the  rate  of  15  mi.  an 
hour,  while  the  current  runs  3  ft.  a  second.     How  long  will  it  take  to 
go  50  mi .? 

3.  A  man  runs  a  quarter  of  a  mile  in  48.4  seconds.     .At  that 
speed,  what  was  his  time  for  100  yd.  ? 

4.  If  a  man  can  run  100  yd.  in  10  sec.,  what  would  be  his  time  for 
a  mile,  if  it  were  possible  to  maintain  the  same  speed  ?     Compare  the 
result  with  the  record  for  a  mile. 


78  MOTION 

5.  A  procession  100  yd.  long,  moving  at  the  rate  of  3  mi.  an  hour, 
passes  over  a  bridge  120  yd.  long.     How  long  does  it  take  the  proces- 
sion to  pass  entirely  over  the  bridge  ? 

6.  An  express  train  is  running  60  mi.  an  hour.     If  the  train  is 
500  ft.  long,  how  many  seconds  will  it  be  in  passing  completely  over  a 
viaduct  160  ft.  in  length? 

7.  A  locomotive  driving  wheel  is  2  m.  in  diameter.     If  it  makes 
200  revolutions  per  minute,  what  is  the  speed  of  the  locomotive  in 
kilometers  per  hour,  assuming  no  slipping  of  the  wheels  on  the  track  ? 

8.  If  a  locomotive  driving  wheel  2  m.  in  diameter  is  making  200 
revolutions  per  minute,  what  is  the  greatest  linear  velocity  of  a  point 
on  its  rim  ?     What  is  the  least  ? 

9.  If  the  acceleration  of  a  marble  rolling  down  an  inclined  plane 
is  40  cm.  per  second  per  second,  what  will  be  its  velocity  after  3  sec. 
from  rest  ? 

10.  How  far  will  a  marble  travel  down  an  inclined  plane  in  3  sec.  if 
the  acceleration  is  40  cm.  per  second  per  second  ? 

11.  A  body  starts  from  rest,  and  moving  with  uniformly  acceler- 
ated motion  acquires   in   10  sec.  a  velocity  of  3600  m.  per  minute. 
What  is  the  acceleration  per  second  per  second.     How  far  does  the 
body  go  in  10  sec.  ? 

12.  What  acceleration  per  minute  per  minute  does  a  body  have  if 
it  starts  from  rest  and  moves  a  distance  of  a  mile  in  5  min.?     What 
will  be  its  velocity  at  the  end  of  4  min.  ? 

13.  If  a  train  acquires  in  2  min.  a  velocity  of  60  mi.  an  hour, 
what  is  its  acceleration  per  minute  per  minute,  assuming  uniformly 
accelerated  motion  ? 

14.  An  electric  car  starting  from  rest  has  uniformly  accelerated 
motion  for  3  min.     At  the  end  of  that  time  its  velocity  is  27  km.  per 
hour.     What  is  its  acceleration  per  minute  per  minute  ? 

15.  A  sled  is  pushed  along  smooth  ice  until  it  has  a  velocity  of  4 
m.  per  second.     It  is  then  released  and  goes  100  m.  before  it  stops. 
If  its  motion  is  uniformly  retarded,  what  is  the  retardation  in  centi- 
meters per  second  per  second  ? 

16.  To  acquire  a  speed  of  60  mi.  an  hour  in  10  min.,  how  far  would 
an  express  train  have  to  run,  provided  it  started  from  rest  and  its  mo- 
tion were  uniformly  accelerated  ? 


CURVILINEAR  MOTION  79 

II.   CURVILINEAR   MOTION 

94.  Direction  of  Motion  on  a  Curve.  —  Curvilinear  motion, 
or  motion  along  a  curved  line,  occurs  more  frequently  in 
nature  than  motion  in  a  straight  line.     The  motion  of  a 
point  on  the  earth's  surface  and  about  its  axis  is  in  a 
circle ;   the  motion  of  the  earth  in  its  path  around  the  sun 
is  along  a  curve  only  approximately  circular  ;    the  motion 
of  a  rocket  or  of  a  stream  of  water 

directed  obliquely  upward  is  along  a 
parabolic  curve,  the  same  as  the  path 
of  many  comets.  So  also  is  the  mo- 
tion of  a  baseball  when  batted  high 
in  air.  The  thrown  "curved  ball," 
too,  illustrates  curvilinear  motion. 

When  the  motion  is  along  a  curved 
line,  the  direction  of  motion  at  any 
point,  as  at  E  (Fig.  80),  is  that  of  Fig*  8' 

the  line  CD,  tangent  to  the  curve  at  the  point.     This  is 
the  same  as  the  direction  of  the  curve  at  the  point. 

95.  Uniform  Circular  Motion.  —  In  uniform  circular  mo- 
tion  the  velocity  of  the  moving   body,  measured  along 
the   circle,  is  constant.     There   is   then   no   acceleration 
in   the   direction   in   which   the   body    is    going   at   any 
point.     But  while  a  velocity  may  remain  unchanged  in 
value,  it   may   vary  in  direction.     If   a   body   is  moving 
with  constant  velocity  in  a  straight  line,  its  acceleration 
is  zero  in  every  direction ;  but  if  the  direction  of  its  motion 
changes  continuously,  then  there  is  an  acceleration  at  right 
angles  to  its  path  and  its  motion  becomes  curvilinear.     If 
this  acceleration  is  constant,  the  motion  is  uniform  in  a 
circle.     Hence,  in  uniform  circular  motion  there  is  a  con- 


80  MOTION 

stant  acceleration  directed  toward  the  center  of  the  circle.     It 
is  called  centripetal  acceleration. 

Uniform  circular  motion  consists  of  a  uniform  motion 
around  the  circumference  of  the  circle  and  a  uniformly 
accelerated  motion  along  the  radius.  If  v  is  the  uniform 
velocity  around  the  circle  whose  radius  is  r,  the  value  of 
the  centripetal  acceleration  is 

O     jit 

a  =  V— ,    .     .     .     .    (Equation  7) 
r 

.  .     .  7         7      _l.         square  of  velocity  in  circle 

or     centripetal  acceleration  =  -4 „  yt — . 

radius  of  circle 

III,    SIMPLE    HARMONIC    MOTION 

96.  Periodic  Motion.  —  The  motion  of  a  body  is  said  to 
be  periodic  when  it  goes  through  the  same  series  of  move- 
ments in  successive  equal  periods  of  time.  If  the  motion 
returns  periodically  to  the  same  value  and  is  as  often  re- 
versed in  direction,  it  is  said  to  be  vibratory.  The  motion 
of  the  earth  around  the  sun  is  periodic,  but  not  vibratory. 
A  hammock  swinging  in  the  wind,  the  pendulum  of  a 

*Let  ABC  (Fig  81)  be  the  circle  in  which  the  body  revolves,  and  AB 
the  minute  portion  of  the  circular  path  described 
in  a  very  small  interval  of  time  t.  Denote  the 
length  of  the  arc  AB  by  s.  Then,  since  the  motion 
along  the  arc  is  uniform,  s  =  vt.  AB  is  the  diago- 
nal of  a  very  small  parallelogram  with  sides  AD 
and  AE.  The  latter  is  the  distance  through  which 
the  revolving  body  is  deflected  toward  the  center 
while  traversing  the  very  small  arc  AB.  Since  the 
acceleration  is  constant,  AE  =  \atz  by  equation 
(6).  The  two  triangles  ABE  and  AB  0  are  simi- 
lar. Hence  AB2  =  AExAC.  Calling  the  radius 
of  the  circle  r  and  substituting  for  AB,  AE,  and  AC  their  values, 

x2r  =  at2r.     Then  a  =  — . 


SIMPLE  HARMONIC  MOTION 


81 


clock,  a  bowed  violin  string,  and  the  prong  of  a  sounding 
tuning  fork  illustrate  both  periodic  and  vibratory  motion. 

97.  Simple  Harmonic  Motion  Described.  —  Simple  harmonic 
motion  is  a  name  given  to  all  pendular  motions  of  small 
amplitude.  The  name  appears  to  be  due 
to  the  fact  that  simple  musical  sounds  are 
caused  by  bodies  vibrating  in  this  manner. 

Suspend  a  ball  by  a  long  thread  and  set  it  swing- 
ing in  a  horizontal  circle  (Fig.  82).  Place  a  white 
screen  back  of  the  ball;  standing  several  feet 
away  and  with  the  eye  E  on  a  level  with  the  ball, 
watch  its 


moving  pro- 
jection on 
the  screen. 
The  eye  dis- 
cerns the 


Fig.  82 


motion  to  the  right  and  to  the  left,  but  not  the  motion  toward  the 
observer  and  away  from  him.  The  apparent  motion  of  the  ball,  as 
viewed  against  the  screen,  is  simple  harmonic. 

Though  the  ball  is  moving  uniformly  around  the  circle,  its  projected 
motion  is  vibratory.  The  velocity  of  this  simple  harmonic  motion  is 
greatest  at  the  middle  of  its  path  A  G,  and  decreases  to  zero  at  either 
extremity. 

Let  the  circle  of  Fig.  83  represent  the  path  of  the  ball, 
and  ABCD,  etc.,  its  projection  on  the  screen.  When  the 
ball  moves  along  the  arc  adg,  it 
appears  to  the  observer  to  move 
from  A  through  B,  C,  etc.,  to  (7, 
where  it  momentarily  comes  to 
rest.  It  then  starts  back  toward 
A,  at  first  very  slowly,  but  with 
increasing  velocity  until  it  passes 
D.  Its  velocity  then  decreases,  and 
at  A  it  is  again  zero,  and  the  motion  Fig.  83 


82  MOTION 

reverses.  At  k  and  d  the  ball  is  moving  across  the  line  of 
sight  and  the  apparent  motion  on  the  screen  is  the  fastest. 
The  radius  of  the  circle,  or  the  distance  AD,  is  the 
amplitude  of  the  vibration.  The  period  of  the  motion  is 
the  time  taken  by  the  ball  to  go  once  around  the  circle ;  it 
is  the  same  as  the  time  of  a  double  oscillation  of  the  pro- 
jected motion.  The  frequency  of  the  vibration  is  the  re- 
ciprocal of  the  period.  For  example,  if  the  period  is  J  a 
second,  the  frequency  is  two  complete  vibrations  a  second. 
This  relation  finds  frequent  illustration  in  musical  sounds, 
where  pitch  depends  on  the  frequency ;  in  light,  where 
frequency  determines  the  color ;  and  in  alternating  cur- 
rents of  electricity,  where  a  frequency  of  50,  for  example, 
means  that  a  complete  wave  is  produced  every  fiftieth  of  a 
second,  and  that  the  current  reverses  100  times  per  second. 

Problems 

-    1.   At  what  speed  must  a  cyclist  ride  around  a  circular  track  one 
inile  in  diameter  in  order  to  go  around  it  in  half  an  hour  ? 

2.  The  circumference  of  the  earth  at  the  equator  is  about  25,000 
mi.      What  is  the  linear  velocity  in  feet  per  second  of  a  point  on  the 
equator,  owing  to  the  earth's  rotation  on  its  axis  ? 

3.  The  diameter  of  the  40th  degree  parallel  of  latitude  is  about 
6000  mi.     What  is  the  speed  in  feet  per  second  of  a  point  on  this 
parallel  on  account  of  the  earth's  rotation  on  its  axis? 

4.  A  flywheel  6  ft.  in  diameter  runs  at  the  rate  of  60  revolutions 
per  minute.     What  is  the  centripetal  acceleration  in  feet  per  second 
per  second  of  a  point  on  the  outer  rim  of  the  wheel  ? 

5.  A  locomotive  driving  wheel  2  m.  in  diameter  makes  200  revo- 
lutions per  minute  on  a  straight  track.     What  is  the  centripetal  ac- 
celeration in  meters  per  second  per  second  for  a  point  at  the  instant 
of  contact  with  the  rail  ? 

6.  A  locomotive  driving  wheel  2  m.  in  diameter  makes  200  revo- 
lutions per  minute  ;  what  is  the  instantaneous  centripetal  acceleration 
for  the  highest  point  of  the  wheel?     (Note.    The  radius  of  rotation 
for  the  highest  point  is  at  that  instant  2  m.) 


CHAPTER  V 
MECHANICS  OF  SOLIDS 
MEASUREMENT  OF  FORCE 

98.  Force. — The  effects  of  force  in  producing  motion 
are  among  our  commonest  experiences.     We  drop  a  knife 
and  it  falls  by  the  force  of  gravity ;  a  mountain  stream 
rushes  downward  by  reason  of  the  same  mysterious  force ; 
the  leaves  of  the  trees  rustle  in  the  breeze,  the  branches 
sway  violently  in  the  wind,  and  their  trunks  are  even 
twisted  off  by  the  force  of  the  tornado ;  powder  explodes 
in  a  rifle  and  the  bullet  speeds  to  its  mark;  loud  thunder 
shakes  the  ground  and  vivid  lightning  rends  a  tree  or 
shatters  a  flagstaff.     From  such  familiar  facts  is  derived 
the  conception  that  force  is  anything  that  produces  motion 
or  change  of  motion  in  material  bodies. 

99.  Units  of  Force.  —  Two  systems  of  measuring  force 
in  common  use  are  the  gravitational  and  the  absolute.     The 
gravitational  unit  of  force  is  the  weight  of  a  standard  mass, 
such  as  the  pound  of  force,  the  gram  of  force,  or  the  kilo- 
gram of  force.     A  pound  of  force  means  one  equal  to  the 
force  required  to  lift  the  mass  of  a  pound  against  the 
downward  pull  of  gravity.     The  same  is  true  of  the  met- 
ric units  with  the  difference  in  the  mass  lifted. 

Gravitational  units  of  force  are  not  strictly  constant 
because  the  weight  of  the  same  mass  varies  from  point  to 
point  on  the  earth's  surface,  and  at  different  elevations. 
The  actual  force  necessary  to  lift  the  mass  of  a  pound  at 

83 


84  MECHANICS  OF  SOLIDS 

the  poles  of  the  earth  is  greater  than  at  the  equator ;  it  is 
less  on  the  top  of  a  high  mountain  than  in  the  neighbor- 
ing valleys,  and  still  less  than  at  the  level  of  the  sea. 
Gravitational  units  of  force  are  convenient  for  the  com- 
mon purposes  of  life  and  for  the  work  of  the  engineer, 
but  they  are  not  suitable  for  precise  measurements. 

The  so-called  " absolute"  unit  of  force  in  the  e.g. s. 
system  is  the  dyne  (from  the  Greek  word  meaning  force} . 
The  dyne  is  the  force  which  imparts  to  a  gram  mass  an 
acceleration  equal  to  one  centimeter  per  second  per  second. 
This  unit  is  invariable  in  value,  for  it  is  independent  of 
the  variable  force  of  gravitation.  It  is  indispensable  in 
framing  the  definitions  of  modern  electrical  and  magnetic 
units. 

100.  Relation  between  the  Gram  of  Force  and  the  Dyne.  — 

The  gram  of  force  is  the  pull  of  the  earth  on  a  mass  of  one 
gram  (the  place  on  the  earth's  surface  is  not  specified). 
Since  the  attraction  of  the  earth  in  New  York  imparts  to 
a  gram  mass  an  acceleration  of  980  cm.  per  second  per 
second,  while  the  dyne  produces  an  acceleration  of  only 
1  cm.  per  second  per  second,  it  follows  that  the  gram  of 
force  in  New  York  is  equal  to  980  dynes,  or  the  dyne  is 
_i_^0f  the  gram  of  force.  The  pull  of  gravity  on  a  gram 
mass  in  other  latitudes  is  not  exactly  the  same  as  in  New 
York,  but  for  the  purposes  of  this  book  it  will  be  suffi- 
ciently accurate  to  say  that  a  gram  of  force  is  equal  to  980 
dynes.  It  will  be  seen,  therefore,  that  the  value  of  any 
force  expressed  in  dynes  is  approximately  980  times  as 
great  as  in  grams  of  force.  Conversely,  to  convert  dynes 
into  grams  of  force,  divide  by  980. 

101.  How  a  Force  is  Measured.  —  The  simplest  device  for 
measuring  a  force  is  the  spring  balance  (Fig.  84).     The 


MEASUREMENT    OF   FORCE  85 

common  draw  scale  is  a  spring  balance  graduated  in  pounds 
and  fractions  of  a  pound.     If  a  weight  of  15  lb.,  for  ex- 
ample, be  hung   on  the  spring  and  the  position 
of  the  pointer  be  marked,  then  any  other  15  lb.     (f"\ 
of  force  will  stretch  the  spring  to  the  same  ex- 
tent in  any  direction.      If    a  man  by  pulling  in 
any  direction  stretches  a  spring   3  in.,  and  if   a 
weight   of   150  pounds  also  stretches  the  spring 
3  in.  the  force  exerted  by  the  man  is  150  pounds 
of  force. 

The  spring  balance  may  be  graduated  in  pounds 
of  force,  kilograms  or  grams  of  force,  or  in  dynes. 
If  correctly  graduated  in  dynes,  it  will  give  right 
readings  at  any  latitude  or  elevation.  Flg*  84 

102.  Graphic  Representation  of  a  Force.  —  A  force  has  not 
only  magnitude  but  also  direction ;  in  addition,  it  is  often 
necessary  to  know  its  point  of  application.  These  three 
particulars  may  be  represented  by  a  straight  line  drawn 

through  the  point  of  applica- 

A~  ~ ~~  s  tion  of  the  force  in  the  direc- 

tion in  which  the  force  acts, 

and  as  many  units  in  length  as  there  are  units  of  force, 
or  some  multiple  or  submultiple  of  that  number.  If  a 
line  1  cm.  long  stands  for  a  force  of  15  dynes,  a  line  4  cm. 
long,  in  the  direction  AB  (Fig.  85),  will  represent  a  force 
of  60  dynes  acting  in  the  direction 
from  A  to  B.  Any  point  on  the  line 
AB  may  be  used  to  indicate  the  point  2kffm\  ^ 
at  which  the  force  is  applied.  A  ~^o 

If  it  is  desired  to  represent  graphi- 
cally the  fact  that  two  forces  act  on  a  body  at  the  same 
time,  for  example,  4  kgm.  of  force  horizontally  and  2  kgm. 


86  MECHANICS  OF  SOLIDS 

of  force  vertically,  two  lines  are  drawn  from  the  point 
of  application  A  (Fig.  86),  one  2  cm.  long  to  the  right, 
and  the  other  1  cm.  long  toward  theJEbp  of  the  page.  The 
lines  AB  and  A  G  represent  the  forces  in  point  of  applica- 
tion, direction,  and  magnitude,  on  a  scale  of  2  kgm.  of 
force  to  the  centimeter. 

II.    COMPOSITION  OF  FORCES  AND  OF  VELOCITIES 

103.  Composition  of   Forces.  —  The   resultant  of  two   or 
more  forces  is  a  single  force  which  will  produce  the  same 
effect  on  the  motion  of  a  body  as  the  several  forces  acting 
together.     (Note  the  exception  in  the  case  of  a  couple, 
§  106.)      The  process  of  finding  the  resultant  of  two  or 
more  forces  is  known  as  the  composition  of  forces.     It  will  be 
convenient  to  consider   first  the  composition  of   parallel 
forces,  and  then  that  of  forces  acting  at  an  angle. 

104.  The  Resultant  of  Parallel  Forces.  —  Suspend  two  draw 
scales,  A  and  B  (Fig.  87),  from  a  suitable  support  by  cords.     Attach 

to  them  a  graduated  bar  supporting 
the  weight  W.  Adjust  the  draw  scales 
and  the  attached  cords  so  that  they 
are  all  vertical.  Read  the  scales  and 
note  the  distances  CE  and  ED ;  also 
compare  the  value  of  W  with  the  sum 
of  the  readings  of  A  and  B.  Change 
the  position  of  the  point  E  and  repeat 
the  observations.  It  will  be  found  in 

each  case  that  —  = Hence  the 

will!  \1  P     CE 

following  principle : 
Fig.  87 

The  resultant  of  two  parallel 

forces  in  the  same  direction  is  equal  to  their  sum  ;  its  point  of 
application  divides  the  line  joining  the  points  of  application  of 
the  two  forces  into  two  parts  which  are  inversely  as  the  forces. 


COMPOSITION  OF  FORCES  AND  OF  VELOCITIES        87 

105.  Equilibrium.  —  It  will  be  apparent  that  the  weight 
IF  is  equal  and  opposite  to  the  resultant  of  the  two  forces 
measured  by  the  draw  scales  A  and  B  (Fig.  87).     The 
three  forces  produce  neither  motion  nor  change  of  motion 
and  are  said  to  be  in  equilibrium;  each  of  the  forces  is 
equal  and  opposite  to  the  resultant  of  the  other  two  and 
is  called  their  equilibrant.     The  equilibrium  of  a  body 
does  not  mean  that  its  velocity  is  zero,  but  that  its  accel- 
eration is  zero.    Rest  means  zero  velocity ;  equilibrium,  zero 
acceleration. 

106.  Parallel   Forces   in    Opposite    Directions.  —  If    two 

parallel  forces,  as  A  and  W  (Fig.  87),  act  in  opposite 
directions,  their  resultant  is  their  difference  (equal  and 
opposite  to  the  force  -B),  and  it  acts  in  the  direction  of 
the  larger  force  (in  this  case  downward  in  the  same  direc- 
tion as  W). 

When  the  two  parallel  forces  acting  in  opposite  direc- 
tions are  equal,  they  form  a  couple.  The  resultant  of  a 
couple  is  zero ;  that  is,  no  single  force  can  be  substituted 
for  it  and  produce  the  same  effect.  A  couple  produces 
motion  of  rotation  only,  in  which  all  the  particles  of  the 
body  to  which  it  is  applied  rotate  in  circles  about  a  com- 
mon axis.  For  example,  a  magnetized  sewing  needle 
floated  on  water  is  acted  on  by  a  couple  when  it  is  dis- 
placed from  a  north-and-south  position.  One  end  of  the 
needle  is  attracted  toward  the  north,  and  the  other  toward 
the  south,  with  equal  and  parallel  forces.  The  effect  is 
to  rotate  the  needle  about  a  vertical  axis  until  it  returns 
to  a  north-and-south  position.  The  common  auger,  as  a 
carpenter  employs  it  to  bore  a  hole,  illustrates  a  couple  in 
the  equal  and  opposite  parallel  forces  applied  by  the  two 
hands. 


88 


MECHANICS   OF  SOLIDS 


107.   The  Resultant  of  Two  Forces  Acting  at  an  Angle.  - 

Tie  together  three  cords  at  D  (Fig.  88)  and  fasten  the  three  ends  to 
the  hooks  of  the  draw  scales  A,  B,  C.  Pass  their  rings  over  pegs  set 
in  a  board  at  such  distances  apart  that  the  draw  scales  will  all  be 
stretched.  Record  the  readings  of  the  scales,  and  by  means  of  a 
protractor  (see  Appendix  I)  measure  the  angles  formed  at  D  by  the 
cords.  Draw  on  a  sheet  of  paper  three  lines  meeting  at  a  point  Z), 
and  forming  with  one  another  these  angles.  Lay  off  on  the  three 


Scale :   SO  gm.  to  1  cm. 


Fig.  88 

lines,  on  some  convenient  scale,  distances  to  represent  the  readings  of 
the  draw  scales,  DP  for  A,  DE  for  B,  and  DC  for  C.  With  DF  and 
DE  as  adjacent  sides,  complete  the  parallelogram  DFGE  and  draw 
the  diagonal  DG.  DG  is  the  resultant  of  the  forces  A  and  B,  and 
its  length  on  the  scale  chosen  will  be  found  equal  to  that  of  Z)C,  their 
equilibrant.  Here  again,  each  force  is  equal  and  opposite  to  the 
resultant  of  the  other  two. 

When  two  forces  act  together  on  a  body  at  an  angle,  the  re- 
sultant lies  between  the  two ;  its  position  and  value  may  be  found 
by  applying  the  following  principle,  known  as  the  parallelogram  of 
forces  : 


COMPOSITION  OF  FORCES  AND   OF  VELOCITIES        89 


If  two  forces  are  represented  by  two  adjacent  sides  (DF 
and  DE)  of  a  parallelogram*  their  resultant  is  represented 
by  the  diagonal  (^DCr)  of  the  parallelogram  drawn  through 
their  common  point  of  application  (-0). 

When  the  two  forces  are  equal,  their  resultant  by  the 
principle  of  symmetry  lies  midway  between  them.  If  the 
two  forces  are  at  right  angles  (Fig.  89), 
the  parallelogram  becomes  a  rectangle 
and  the  two  forces  and  their  resultant 
are  represented  by  the  three  sides  of  a 
right  triangle,  AB,  BD,  AD.  The 
value  of  the  resultant  in  this  case  may 


Fig.  89 


be  found  by  computing  the  hypotenuse  of  the  triangle. 

For  example,  if  the  forces  at  right  angles  are  6  kgm.  of  force  and 
8  kgm.  of  force,  their  resultant  is 

V62  +  82  =  10  kgm.  of  force. 

108.  Component  of  a  Force  in  a  Given  Direction.  — 
frequently  occurs  that  if  a  force  produces  any  motion, 
must  be  in  a  direction  other  than  that  of  the  force  itsel 

For     example,     suppose 
force  AB  (Fig.    90)  appl 
to  cause  a  car  to  move  alon 
the  rails  mn.     The  force 
evidently    produces    tw 


Fig.  90 


effects ;  it  tends  to  move  the  car  along  the  rails,  and  it  in- 
creases the  pressure  on  them.  The  two  effects  are  pro- 
duced by  the  two  forces  OB  and  DB  respectively.  They 
are  therefore  the  equivalent  of  AB.  The  force  OB  is 
called  the  component  of  AB  in  the  direction  of  the  rails 
mn,  and  DB  is  the  component  perpendicular  to  them. 
The  component  of  a  force  in  a  given  direction  is  its  effective 
value  in  this  direction. 

To  find  the  component  in  a  given  direction,  construct  on 


90 


MECHANICS  OF  SOLIDS 


the  line  representing  the  force,  as  the.  diagonal,  a  rectangle, 
the  sides  of  which  are  respectively  parallel  and  perpendicular 
to  the  direction  of  the  required  component  ;  the  length  of  the 
side  parallel  to  the  given  direction  represent*  the  component 
sought. 

EXAMPLE.  Let  a  force  of  200  Ib.  be  applied  to  a  truck,  as  AB  in 
Fig.  90  ;  and  let  it  act  at  an  angle  of  30°  with  the  horizontal.  Find 
the  horizontal  component  pushing  the  truck  forward. 

Construct  a  parallelogram  (see  Appendix)  with  the  angle  ABC 
equal  to  30°  and  measure  the  side  CB.  Or,  remembering  that  the 
side  opposite  an  angle  of  30°  is  half  the  hypotenuse,  it  follows  that 
AC  is  100  Ib.  of  force,  and  the.  third  side  of  the  right  triangle  is  then 


CB  = 


-  1002  =  173.2  Ib.  of  force. 


The  kite  and  the  sailboat  are  two  familiar  illustrations  of  the  prin- 

ciple.    In  the  case  of  the  kite,  the  forces  acting  are  the  weight  of  the 

kite  AB  (Fig.  91),  the  pull  of  the 
string  AC,  and  the  force  of  the 
wind.  The  force  of  the  wind  may 
be  resolved  into  two-  components, 
one  perpendicular  to  the  face  of  the 
kite  HK,  and  the  other  parallel 
to  the  face.  If  the  component  per- 
pendicular to  the 
face  equals  AD, 
the  resultant  of 
AB  and  AC,  then 
the  kite  will  be  at 
rest  ;  if  greater, 

then  the  kite  will  move  upward  ;  if  less,  the  kite 

will  descend. 

In  the  case  of  the  sailboat,  the  sail  is  set  at  such 

an  angle  that  the  wind  strikes  the  rear  face.     In 

Fig.  92,  BS  represents  the  sail,  and  AB  the  direc- 

tion and  force  of  the  wind.     This  force  may  be  re- 

solved into  two  rectangular  components,  CB  and 

DB,  of  which  CB  represents  the  intensity  of  the 

force  that  drives  the  boat  forward. 


Fig.  92 


COMPOSITION   OF  FORCES  AND   OF  VELOCITIES        91 

109.  Composition  and  Resolution  of  Velocities.  —  At  the 
Paris  exposition  in  1900  a  continuous  moving  sidewalk 
carried  visitors  around  the  grounds.  A  person  walking 
on  this  platform  had  a  velocity  with  respect  to  the  ground 
made  up  of  the  velocity  of  the  sidewalk  relative  to  the 
ground  and  the  velocity  of  the  person  relative  to  the  mov- 
ing walk.  The  several  velocities  entering  the  result  are 
the  component  velocities.  Velocities  may  be  combined  and 
resolved  by  the  same  methods  as  those  applying  to  forces. 
When  several  motions  are  given  to  a  body  at  the  same 
time,  its  actual  motion  is  a  compromise  between  them,  and 
the  compromise  path  is  the  resultant. 

The  following  is  an  example  of  the  composition  of  two  velocities 
at  right  angles  :  A  boat  can  be  rowed  in  still  water  at  the  rate  of  5 
mi.  an  hour  ;  what  will  be  its  actual  velocity  if  it  be  rowed  5  mi. 
an  hour  across  a  stream  running  3  mi.  an  hour? 

Let  AB  (Fig.  93)  represent  in  length  and  direction  the  velocity  of 
5  mi.  an  hour  across  the  stream,  and  AC,  at  right  angles  to  AB,  the 
velocity  of  the  current,  3  mi.  an  hour, 
both  on  the  same  scale.  Complete  the 
parallelogram  ABDC,  and  draw  the  diago- 
nal AD  through  the  point  A  common  to 
the  two  component  velocities.  A  D  repre- 
sents the  actual  velocity  of  the  boat;  its 
length  on  the  same  scale  as  that  of  the 
other  lines  is  5.83:  The  resultant  velocity  Fig<  93 

is  therefore  5.83  miles  an  hour  in  the  direction  AD. 

When  the  angle  between  the  components  is  aright  angle,  as  in  the 
present  case,  the  diagonal  AD  is  the  hypotenuse  of  the  right  triangle 
ABD.  Its  square  is  therefore  the  sum  of  the  squares  of  5  and  3,  or 


When  the  angle  at  A  is  not  a  right  angle,  the  approximate  resultant 
may  be  found  by  a  graphic  process  of  measurement. 

A  velocity,  like  a  force,  has  both  direction  and  magni- 


92  MECHANICS  OF  SOLIDS 

tude,  and  a  component  of  it  in  any  given  direction  may  be 
found  in  precisely  the  same  way  as  in  the  case  of  a  force, 

§108. 

Problems 

NOTE.   Solve  graphically  the  problems  involving  forces  or  velocities  at  an 
angle.    Consult  Appendix  I  for  methods  of  drawing. 

1.  A  body  is  acted  on  by  two  parallel  forces  of  20  and  30  Ib.  of 
force  in  the  same  direction  ;  their  points  of  application  are  60  in.  apart. 
Find  the  value  of  the  resultant  and  the  distance  of  its  point  of  appli- 
cation from  either  force. 

SUGGESTION.  Let  x  be  the  distance  of  the  point  of  application  of  the  re- 
sultant from  that  of  the  force  20.    Then  60  —  x  is  its  distance  from  the  point 

of  application  of  the  force  30,  and  by  §104,  —  =  60~x . 

oO  X 

2.  A  weight  of  210  Ib.  is  carried  at  the  middle  of  a  bar  6  ft.  long 
by  a  boy  at  one  end  and  a  man   at  the  other  side  of   the   middle. 
Where  must  the  man  take  hold  so  that  he  shall  carry  twice  as  much 
as  the  boy  ? 

3.  A  horse  and  a  colt  are  hitched  side  by  side  to  a  loaded  wagon. 
At  what  point  of  the  double  tree  must  it  be  attached  to  the  tongue  of 
the  wagon  so  that  the  colt  will  pull  two  pounds  to  three  pounds  of 
force  pulled  by  the  horse,  the  doubletree  being  40  in.  long? 

4.  Three  parallel  forces  are  acting  on  a  bar  104  cm.  long.     Two 
of  them,  500   and   800   dynes  respectively,  are  applied  at  the  ends. 
What  must  be  the  third  force  and  where  must  it  be  applied  so  that 
the  bar  may  remain  at  rest  and  the  three  forces  be  in  equilibrium  ? 

5.  Resolve  a  force  of  100  dynes  into  two  parallel  forces  with  their 
points  of  application  20  and  30  cm.,  respectively,  from  that  of  the 
original  force. 

6.  Two  parallel  forces  of  100  and  150  dynes,  respectively,  have 
their  points  of  application  50  cm.  apart.     What  third  parallel  force 
will  produce  equilibrium,  and  where  must  it  be  applied  so  that  the 
three  points  of  application  are  in  the  same  straight  line  ? 

7.  Two  forces,  30  and  40  grams  of  force,  act  at  an  angle  of  60°. 
Find  the  resultant. 

8.  A  train  is  running  with  a  speed  of  30  mi.  an  hour.     A  package 
is  thrown  perpendicularly  from  it  with  a  velocity  of  20  ft.  a  second. 
What  is  the  velocity  of  the  package  with  respect  to  the  ground? 


NEWTON'S  LAWS  OF  MOTION 


93 


9.  A  sailboat  is  going  eastward,  the  wind  is  from  the  northwest, 
and  the  sail  is  set  at  an  angle  of  30°  with  the  direction  of  the  wind. 
If  the  wind  is  blowing  12  mi.  an  hour,  what  is  its  component  perpen- 
dicular to  the  sail  ? 

10.  A  mass  of  30  gm.  is  suspended  by  a  string. 
It  is  pulled  aside  by  a  horizontal  force  of  17.32  gin. 
of  force,  and  the  string  then  makes  an  angle  of  30° 
with  the  vertical  (Fig.  94).     Find  the  tension  in 
the  string.     (Note  that  in  a  right  triangle  with  a 
30°  angle,  the  side  opposite  this  angle  is  half  the 
hypotenuse.) 

11.  Horses  attached  to  a  car  pull  at  an  angle  of 
30°  with  the  track  and  with  a  force  of  1200  Ib. 
What  is  the  force  moving  the  car? 

12.  A  cord  supporting  a  picture  weighing  15  Ib. 

passes  over  a  knob  so  that  the  angle  between  the  two  parts  of  the 
cord  is  60°.     What  is  the  tension  in  the  cord  ? 


III.    NEWTON'S  LAWS   OF  MOTION 

110.  Momentum.  —  So  far  we  have  considered  motion 
without  reference  to  the  mass  moved,  and  without  con- 
sidering the  relation  between  force  on  the  one  hand  and 
the  moving  mass  and  its  velocity  on  the  other.  Before 
taking  up  the  laws  of  motion,  which  outline  the  relations 
between  force  and  motion,  it  is  necessary  to  define  two 
terms  intimately  associated  with  these  laws.  The  first  of 
these  is  momentum.  Momentum  is  the  product  of  the  mass 
and1  the  linear  velocity  of  a  moving  body. 

Momentum  =  mass  x  velocity,  or  M  =  mv.     (Equation  8) 

In  the  e.g.  s.  system,  the  unit  of  momentum  is  the  mo- 
mentum of  a  mass  of  1  gm.  moving  with  a  velocity  of  1  cm. 
per  second.  It  has  no  recognized  name.  In  the  English 
system,  the  unit  of  momentum  is  the  momentum  of  a  mass 
of  1  Ib.  moving  with  a  velocity  of  1  ft.  per  second. 


94  MECHANICS   OF  SOLIDS 

111.  Impulse.  —  Suppose  a  ball  of  10  gm.   mass  to  be 
fired  from  a  rifle  with  a  velocity  of  50,000  cm.  per  second. 
Its  momentum  would  be  500,000  units.     If  a  truck  weigh- 
ing $0  kgm.  moves. .at  the  rate  of  10  cm.  per  second,  its 
momentum  is  also  500,000  units.     But  the  ball  has  acquired 
its  momentum  in  a  fraction  of  a  second,  while  a  minute 
or  more  may  have  been  spent  in  giving  to  the  truck  the 
same  momentum.     In  some  sense  the  effort  required  to 
set  the  ball  in  motion  is  the  same  as  that  required  to  give 
the  equivalent  amount  of  motion  to  the  truck,  because  the 
momenta  of  the  two  are  equal. 

*  This  equality  is  expressed  by  saying  that  the  impulse 
is  the  same  in  the  two  cases.  Since  the  effect  is  doubled 
if  the  value  of  the  force  is  doubled,  or  if  the  time  during 
which  the  force  continues  to  act  is  doubled,  it  follows 
that  impulse  is  the  product  of  the  force  and  the  time  it  con- 
tinues to  act.  In  estimating  the  effect  of  a  force,  the  time 
element  and  the  magnitude  of  the  force  are  equally  im- 
portant. The  term  impulse  takes  both  into  account. 

112.  The  Laws  of  Motion. — The  laws  of  motion,  first 
enunciated    by    Sir  Isaac    Newton,  are    to    be   regarded 
as  physical   axioms,  incapable  of   rigorous  experimental 
proof.    They  must  be  considered  as  resting  on  convictions 
drawn  from  observations  and  experiment  in  the  domain 
of   physics   and    astronomy.      The    results   derived  from 
their  application  have  so  far  been  found  to  be  invariably 
true.     They  may  be  stated  as  follows : 

I.  Every  body  continues  in  Us  state  of  rest  or  of  uni- 
form motion  in  a  straight  line,  except  in  so  far  as  it  may 
be  compelled  by  applied  force  to  change  that  state. 

II.  Change  of  momentum  is  proportional  to  the  im- 
pulse which  produces  it,  and  takes  place  in  the  direction 
in  which  the  force  acts. 


Sir  Isaac  Newton  (1642-1727)  is  celebrated  for  his  discoveries 
in  mathematics  and  physics.  He  was  a  Fellow  of  Trinity  Col- 
lege, Cambridge.  He  discovered  the  binomial  theorem  in  alge- 
bra and  laid  the  foundation  of  the  calculus.  His  greatest  work  is 
the  Prmcipia,  a  treatise  on  motion  and  the  laws  governing  it.  His 
greatest  discoveries  are  the  laws  of  gravitation  and  the  composi- 
tion of  white  light. 

From  Kepler's  laws  of  the  planetary  orbits  Newton  proved  that 
the  attraction  of  the  sun  on  the  planets  varies  inversely  as  the 
squares  of  their  distances. 

He  was  also  distinguished  in  public  life.  He  sat  in  Parliament 
for  the  University  of  Cambridge,  was  at  one  time  Master  of  the 
Mint,  and  the  reformation  of  the  English  coinage  was  largely  his 
work. 


NEWTON'S  LAWS  OF  MOTION  95 

III.  To  every  action  there  is  always  an  equal  and  con- 
trary reaction;  or  the  mutual  actions  of  two  bodies  are 
always  equal  and  oppositely  directed. 

113.  Discussion  of  the  First  Law.  —  This  law  is  known  as 
the  law  of  inertia  (§  7),  since  it  asserts  that  a  body  per- 
sists in  its  condition  of  either  rest  or  uniform   motion, 
unless  it  is  compelled  to  change  that  state  by  the  action 
of  an  external  force.     It  is  further  true  that  a  body  offers 
resistance  to  any  such  change  in  proportion  to  its  mass. 
Hence  the  term  mass  is  now  commonly  used  to  denote  the 
measure  of  the  body's  inertia  (§  9). 

From  this  law  we  also  derive  the  Newtonian  definition 
of  force,  for  the  law  asserts  that  force  is  the  sole  cause  of 
change  of  motion. 

114.  Discussion  of  the  Second  Law.  —  The  first  law  teaches 
that  a  change  of  motion  is  due  to  impressed  force.     The 
second  law  points  out,  in  the  first  place,  what  the  measure 
of  a  force  is.    It  was  restated  by  Maxwell  as  follows  :  "  The 
change  of  momentum  of  a  body  is  numerically  equal  to  the 
impulse  which  produces  it,  and  is  in  the  same  direction,"  or 

momentum  (mass  x  velocity)  =  impulse  (force  x  time). 
Expressed  in  symbols,      mv—ft.     .     .     .     (Equation  9) 

Hence,  /=^. 

t 

The  initial  velocity  of  the  mass  m  before  the  force  / 
acted  on  it  is  here  assumed  to  be  zero,  and  v  is  the  veloc- 
ity attained  in  t  seconds.  Then  the  total  momentum  im- 
parted in  the  time  t  is  mv,  and  therefore  -  -  is  the  rate  of 

C 

change  of  momentum.     Force  is  therefore  measured  by  the 

rate  of  change  of  momentum.     Since  -  is  the  rate  of  change 

t 


96  MECHANICS  OF  SOLIDS 

of  velocity,  or   the  acceleration  a  (see  equation    5),  we 
may  write 

f=ma.      .     .     .     (Equation  10) 

We  see  from  this  that  force  may  also  be  measured  by  the 
product  of  the  mass  moved  and  the  acceleration  imparted  to 
it.  Therefore  when  the  mass  m  is  unity,  the  force  is 
numerically  equal  to  the  acceleration  it  produces.  Hence 
the  definition  of  the  dyne  (§  99). 

This  law  teaches,  in  the  second  place,  that  the  change  of 
momentum  is  always  in  the  direction  in  which  the  force  acts. 
Hence,  when  two  or  more  forces  act  together,  each  pro- 
duces its  change  of  momentum  independently  of  the  others 
and  in  its  own  direction.  This  principle  lies  at  the  founda- 
tion of  the  method  of  finding  the  resultant  effect  of  two 
forces  acting  on  a  body  in  different  directions  (§  107). 

On   a  horizontal   shelf    about  two 
meters   above  the   floor  are  placed  two 
marbles,  one  on  each  side  of  a  straight  spring 
fixed    vertically   over    a   hole    in    the   shelf.      One 
marble  rests  on  the  shelf  and  the  other  is  held  over 
the  hole  between  the  spring  and  a  block  fixed  to  the 
shelf  (Fig.  95).     When  the  hammer  falls  and  strikes 
the  spring,  it  projects  the  one  marble  horizontally 
and  lets  the  other  one  fall  vertically.     The  two  reach  the  floor  at  the 
same  instant.     Both  marbles  have  the  same  vertical  acceleration. 

115.  Discussion  of  the  Third  Law.  —  The  essence  of  this 
law  is  that  all  action  between  two  bodies  is  mutual.  Such 
action  is  known  as  a  stress  and  a  stress  is  always  a  two- 
sided  phenomenon,  including  both  action  and  reaction. 
The  third  law  teaches  that  these  two  aspects  of  a  stress 
are  always  equal  and  in  opposite  directions.  The  stress 
in  a  stretched  elastic  cord  pulls  the  two  bodies  to  which  it 
is  attached  equally  in  opposite  directions ;  the  stress  in  a 


NEWTON'S  LAWS  OF  MOTION  97 

compressed  rubber  buffer  or  spring  exerts  an  equal  push 
both  ways ;  the  former  is  called  a  tension  and  the  latter  a 
pressure. 

ILLUSTRATIONS.  The  tension  in  a  rope  supporting  a  weight  is  a 
stress  tending  to  part  it  by  pulling  adjacent  portions  in  opposite  di- 
rections. The  same  is  obviously  true  if  two  men  pull  at  the  ends  of 
the  rope.  An  ocean  steamship  is  pushed  along  by  the  reaction  of  the 
water  against  the  blades  of  the  propeller.  The  same  is  true  of  an 
aeroplane,  only  in  this  case  the  reaction  against  the  blades  is  by  the 
air,  and  the  blades  are  longer  and  revolve  much  faster  than  in  water 
in  order  to  move  enough  air  to  furnish  the  necessary  reaction.  When 
a  man  jumps  from  a  rowboat  to  the  shore,  he  thrusts  the  boat  back- 
wards. An  athlete  would  not  make  a  record  standing  jump  from  a 
feather  bed  or  a  spring  board.  When  a  ball  is  shot  from  a  gun,  the 
gun  recoils  or  "  kicks."  All  attraction,  such  as  that  between  a  mag- 
net and  a  piece  of  iron,  is  a  stress,  the  magnet  attracting  the  iron  and 
the  iron  the  magnet  with  the  same  force. 

Practical  use  is  made  of  reaction  to  turn  the  oscillating  electric  fan 
from  side  to  side  so  as  to  blow  the  air  in  different  directions.  A  rec- 
tangular sheet  of  brass  is  bent  lengthwise  at 
right  angles  and  is  pivoted  so  as  to  turn  90° 
about  a  vertical  axis  (Fig.  96).  When  one 
half  of  this  bent  sheet  is  exposed  to  the  air  cur- 
rent, the  reaction  sustained  by  the  blades  of 
the  fan  on  this  side  is  in  part  balanced  by  the 
reaction  of  the  bent  sheet ;  but  on  the  opposite 
half  of  the  fan  the  reaction  of  the  blades  is 
not  balanced.  Hence  the  whole  fan  turns 
about  a  vertical  axis  on  the  standard  until  a 
lever  touches  a  stop  and  shifts  the  bent  strip  Fig.  96 

so  as  to  expose  the  other  half  of  it  to  the  air 

current  from  the  opposite  half  of  the  fan.  The  fan  then  reverses  its 
slow  motion  and  turns  to  the  other  side. 

Since  force  is  measured  by  the  rate  at  which  momentum 
changes,  the  third  law  of  motion  is  equivalent  to  the  fol- 
lowing : 

In  every  action  between  two  bodies,  the  momentum 


98  MECHANICS  OF  SOLIDS 

gained  by  the  one  is  equal  to  that  lost  by  the  other,  or  the 
momenta  in  opposite  directions  are  the  same. 

Problems 

1.  Why  will  not  the  same  impulse  impart  to  a  2-lb.  ball  the  same 
velocity  as  it  does  to  a  1-lb.  ball  ? 

2.  What  is  the  momentum  of  a  mass  of  250  gm.  moving  with  a 
velocity  of  75  cm.  per  second  ? 

3.  Calculate  the  ratio  of  the  momentum  of  a  ball  whose  mass  is 
5  Ib.  and  speed  1500  ft.  per  second  to  that  of  a  50-lb.  ball  moving 
with  a  speed  of  1800  ft.  per  minute. 

4.  What  velocity  must  be  given  to  a  mass  of  25  kgm.  that  it  may 
have  the  same  momentum  as  a  75-kgm.  ball  moving  with  a  velocity  of 
1500  cm.  per  second  ? 

5.  A  10-lb.  gun  is  loaded  with  a  half-ounce  ball.      When  fired 
the  ball  has  a  velocity  of  1800  ft.  per  second.     What  is  the  velocity  of 
recoil  of  the  gun  ? 

6.  Two  balls  have  equal  momenta.     The  first  has  a  mass  of  200 
gm.  and  a  velocity  of  20  m.  per  second ;  the  second  has  a  velocity  of 
600  m.  per  minute ;  what  is  its  mass  ? 

7.  An  unbalanced  force  acts  on  a  mass  of  100  gm.  for  5  seconds, 
and  imparts  to  it  a  velocity  of  2.5  cm.  per  second.     What  is  the  value 
of  the  force  in  dynes  (§  99)  ? 

8.  What  is  the  acceleration  when  a  force  of  50  dynes  acts  on  a 
mass  of  8  gm.?     How  far  will  the  mass  of  8  gm.  move  in  4  sec.  V 

9.  An  unbalanced  force  of  60  dynes  acts  on  a  body  for  1  min. 
and  gives  to  it  a  velocity  of  1200  cm.  per  second.     What  is  the  mass 
of  the  body  ? 

10.  Two  grams  of  force  act  continuously  on  a  mass  of  49  gm.  for 
10  seconds.  Find  the  velocity  acquired  and  the  space  traversed. 
(To  apply  equations  (9)  and  (10)  the  force  must  be  expressed  in 
dynes.) 

IV.     GRAVITATION 

116.  Weight.  —  The  attraction  of  the  earth  for  all  bodies  is 
called  gravity.  The  weight  of  a  body  is  the  measure  of  this 


GRAVITATION  99 

attraction.  It  is  a  pull  on  the  body  and  therefore  a  force. 
It  makes  a  body  fall  with  uniform  acceleration  called 
the  acceleration  of  gravity  #nd  denoted  by  g.  If 
we  represent  the  weight  of  a  body  by  w  and  its  mass 
by  m,  by  equation  (10),  w  =  mg.  From  this  it  ap- 
pears that  the  weight  of  a  body  is  proportional  to  its 
mass,  and  that  the  ratio  of  the  weights  of  two  bodies 
at  any  place  is  the  same  as  that  of  their  masses. 
Hence,  in  the  process  of  weighing  with  a  beam 
balance,  the  mass  of  the  body  weighed  is  compared 
with  that  of  a  standard  mass.  When  a  beam  bal- 
ance shows  equality  of  weights,  it  shows  also  equal- 
ity of  masses. 

117.  Direction    of    Gravity.  —  The    direction    in 
which  the  force  of  gravity  acts  at  any  point  is  very 
nearly  toward  the  earth's  center.     It  may  be  deter- 
mined by  suspending  a  weight  by  a  cord  passing 
through  the  point.     The  cord  is  called  a  plumb  line 
(Fig.  97),  and  its  direction  is  a  vertical  line.     A    fjf 
plane  or  line  perpendicular  to  a  plumb  line  is  said  p.    97 
to   be    horizontal.     Vertical   lines    drawn    through 
neighboring  points   may  be    considered  parallel  without 
sensible  error. 

118.  Center  of   Gravity. — A  body  is   conceived  to  be 
composed  of  an  indefinitely  large  number  of  particles,  each 
of  which  is  acted  on  by  gravity.     For  bodies  of  ordinary 
size,  these  forces  of  gravity  are  parallel  and  proportional 
to  the  masses  of  the  several  small    parts.      The  point  of 
application  of  their  resultant  is  the  center  of  gravity  of  the 
body. 

If  the  body  is  uniform  throughout,  the  position  of  its 


100  MECHANICS  OF   SOLIDS 

center  of  gravity  depends  on  its  geometrical  figure  only. 
Thus,  the  center  of  gravity  (1)  of  a  straight  rod  is  its 
middle  point ;  (2)  of  a  circle  or  ring,  its  center  ;  (3)  of  a 
sphere  or  a  spherical  shell,  its  center  ;  (4)  of  a  parallelo- 
gram, the'  intersection  of  its  diagonals  ;  (5)  of  a  cylinder 
or  a  cylindrical  pipe,  the  middle  point  of  its  axis. 

It  is  necessary  to  guard  against  the  idea  that  the  force 
of  gravity  on  a  body  acts  at  its  center  of  gravity. 
Gravity  acts  on  all  the  particles  composing  the  body,  but 
its  effect  is  generally  the  same  as  if  the  resultant,  that  is, 
the  weight  of  the  body,  acted  at  its  center  of-  gravity.  It 
will  be  seen  from  the  examples"  of  the  ring  and  the  cylin- 
drical pipe  that  the  center  of  gravity  may  lie  entirely  out- 
side the  body. 

119.  Law  of  Universal  Gravitation.  —  It  had  occurred  to 
Galileo  and  other  early  philosophers  that  the  attraction 
of  gravity  extends  beyond  the  earth's  surface,  but  it  re- 
mained for  Sir  Isaac  Newton  to  discover  the  law  of  uni- 
versal gravitation.  He  derived  this  great  generalization 
from  a  study  of  the  planetary  motions  discovered  by  Kep- 
ler. The  law  may  be  expressed  as  follows: 

Every  portion  of  matter  in  the  universe  attracts  every 
other  portion,  and  the  stress  between  them  is  directly  pro- 
portional to  the  product  of  their  masses  and  inversely  pro- 
portional to  the  square  of  the  distance  between  them. 

For  spherical  bodies,  like  the  sun,  the  earth,  and  the 
planets,  the  attraction  of  gravitation  is  the  same  as  if  all 
the  matter  in  them  were  concentrated  at  their  centers  ; 
hence,  in  applying  to  them  the  law  of  gravitation,  the  dis- 
tance between  them  is  the  distance  between  their  centers. 
Calculationsunade  to  find  the  centripetal  acceleration  of 
the  moon  in  its  orbit  show  that  it  is  attracted  to  the  earth 


GRAVITATION  101 

with  a  force  which  follows  the  law  of  universal  gravita- 
tion.1 

120.  Variation  of  Weight.  —  Since  the  earth  is  flattened  at 
the  poles,  it  follows  from  the  law  of  gravitation  that  the  ac- 
celeration of  gravity,  and  the  weight  of  any  body,  increase 
in  going  from  the  equator  toward  the  poles.     If  the  earth 
were  a  uniform  sphere  and  stationary,  the  value  of  g  would 
be  the  same  all  over  its  surface.     But  the  value  of  g  varies 
from  point  to  point  on  the  earth's  surface,  even  at  sea  level, 
both  because  the  earth  is  not  a  sphere  and  because  it  rotates 
on  its  axis.     The  centripetal  acceleration  of  a  point  at  the 
equator,  owing  to  the  earth's  rotation  on  its  axis,  is  5- j-^ 
the  acceleration  of  gravity  g.     Since  289  is  the  square  of 
17,  and  the 'centripetal  acceleration  varies  as  the  square  of 
the  velocity  (§95),  it  follows  that  if  the  earth  were  to  rotate 
in  one  seventeenth  of  a  day,  that  is,  17  times  as  fast  as  it 
now  rotates,  the  apparent  value  of  g  at  the  equator  would 
become  zero,  and  bodies  there  would  lose  all  their  weight. 

The  value  of  g  at  the  equator  is  978.1  and  at  the  poles 
983.1,  both  in  centimeters  per  second  per  second.  At 
New  York  it  is  980.15  centimeters,  or  32.16  feet,  per  sec- 
ond per  second. 

121.  Equilibrium  under  Gravity.  —  When  a  body  rests  on 
a  horizontal  plane,  its  weight  is  equal  and  opposite  to  the 
reaction  of  the  plane.     The  vertical  line  through  its  center 
of  gravity  must  therefore  fall  within  its  base  of  support. 
If  this  vertical  line  falls  outside  the  base,  the  weight  of 
the  body  and  the  reaction  of  the  plane  form  a  couple 
(§  106),  and  the  body  overturns. 

The  three  kinds  of  equilibrium  are,  (1)  stable,  for  any 
displacement  which  causes  the  center  of  gravity  to  rise ; 

iCarhart's  College  Physics,  §  58. 


102 


MECHANICS  OF  SOLIDS 


(2)  unstable,  for  any  displacement  which  causes  the  center 
of  gravity  to  fall ;  (3)  neutral,  for  any  displacement  which 
does  not  change  the  height  of  the  center  of  gravity. 

Fill  a  round-bottomed  Florence  flask  one  quarter  full  of  shot  and 
cover  them  with  melted  paraffin  to  keep  them  in  place  (Fig.  98).     Tip 

the  flask  over  ;  after  a  few  os- 
cillations it  will  return  to  an 
upright  position.  Repeat  the 
experiment  with  a  similar 
empty  flask ;  it  will  not  stand 
up,  but  will  rest  in  any  posi- 
tion on  its  side  and  with  the 
top  on  the  table.  The  loaded 
flask  cannot  be  tilted  over 


Fig  98 


without  raising  its  center  of  gravity ;  in  a  vertical  position  it  is  there- 
fore stable  and  when  tipped  over,  unstable,  for  it  returns  to  a  ver- 
tical position.  For  the  empty 
flask,  its  center  of  gravity  is 
lower  when  it  lies  on  its  side 
than  when  it  is  erect.  Rolling 
it  around  does  not  change  the 
height  of  its  center  of  gravity 
and  its  equilibrium  is  thus 
neutral. 

The  three  funnels  of  Fig.  99 


Fig.  99 


illustrate  the  three  kinds  of  equilibrium  on  a  plane. 

A  rocking  horse,  a  rocking  chair,  and  a  half  sphere  resting  on  its 
convex  side  are  examples  of  stable  equilibrium. 
An  egg  lying  on  its  side  is  in  neutral  equilibrium 
for  rolling  and  stable  equilibrium  for  rocking ;  it  is 
unstable  on  either  end.  A  lead  pencil  supported 
on  its  point  is  in  unstable  equilibrium.  Any  such 
body  may  become  stable  by  attaching  weights  to 
it  in  such  a  manner  as  to  lower  the  center  of  grav- 
Fig.  100  ity  below  the  supporting  point  (Fig.  100). 

122.    Stability.  —  Stability   is   the  state  of  being  firm  or 
stable.    The  higher  the  center  of  gravity  of  a  body  must  be 


QUESTIONS  AND  PROBLEMS 


103 


lifted  to  put  the  body  in  unstable  equilibrium  or  to  over- 
turn it,  the  greater  is  its  stability.  This  condition  is  met 
by  a  relatively  large  base  and  a  low  center  of  gravity.  A 
pyramid  is  a  very  stable  form.  On  account  of  the  large  area 
lying  within  the  four  feet  of  a  quadruped,  its  stability  is 
greater  than  that  of  a  biped.  A  child  is  therefore  able  to 
creep  "  on  all  fours  "  before  it  learns  to  maintain  stable 
equilibrium  in  walking.  A  boy  on  stilts  has  smaller  sta- 
bility than  on  his  feet  because  his  support  is  smaller  and 
his  center  of  gravity  higher. 

Stability  may  be  well  illustrated  by  means  of  a  brick.  It  has 
greater  stability  when  lying  on  its  narrow  side  (2"  x  8")  than  when 

_^  standing  on  end;  and  on  its 

><      broad  side  (4"  x  8")  its  sta- 
B  ,'  1     bility  is    still    greater.     Let 

/        Fig.  101  represent  a  brick  ly-; 
/          ing  on  its  narrow  side  in  A 
and  standing  on  end  in  B. 
In  both  cases  to  overturn  it 
Fig.  1 01  its  center  of  gravity  c  is  lifted 

to   the  same  height,  but  the 

vertical  distance  bd  through  which  the  center  of  gravity  must  be  lifted 
is  greater  in  A  than  in  B. 

A  tall  chimney  or  tower  has  no  great  stability  because  its  base  is 
relatively  small  and  its  center  of  gravity  high.  A  high  brick  wall  is 
able  to  support  a  great  crushing  weight,  but  its  stability  is  small  unless 
it  is  held  by  lateral  walls  and  floor  beams. 

Questions  and  Problems 

1.  If  one  jumps  off  the  top  of  an 
empty  barrel  standing  on  end,  why  is 
one  likely  to  get  a  fall  ? 

2.  Where  is  the  center  of  gravity 
of  a  knife  supported  as  in  Fig.  102  ? 

3.  Can  you  devise  a  method  of  find- 
ing where  the  center  of  gravity  of  a  uniform  triangle  is  ? 


Fig.  102 


104  MECHANICS   OF  SOLIDS 

4.  Show  by  a  figure  why  a  ball  rolls  down  hill  the  faster,  the 
steeper  the  hill. 

5.  Why  are  low  wagons  better  adapted  for  use  on  hillsides  than 
high  ones  ? 

6.  If  a  body  at  the  equator  weighs  9781  gm.  on  a  spring  balance, 
what  would  it  weigh  at  the  north  pole  ?     (Weight  is  proportional  to 
<70 

7.  If  a  body  weighs  16  kgm.  at  sea  level  on  a  spring  balance,  how 
many  grams  less  will  it  weigh  on  the  top  of  a  mountain  2  mi.  above 
sea  level,  if  the  radius  of  the  earth  is  4000  mi.?     (The  value  of  g 
is  inversely  as  the  square  of  the  distance  from  the  center  of  the 
earth.) 

8.  If  the  acceleration  of  gravity  g  is  32.2  ft.  per  second  per  second 
on  the  earth's  surface,  what  is  it  at  a  distance  equal  to  that  of  the 
moon,  if  the  earth's  radius  is  4000  mi.  and  the  distance  of  the  moon 
240,000  mi.  ? 

9.  If  the  acceleration  of  gravity  on  the  earth  is  980  cm.  per  second 
per  second,  what  is  it  on  Mars,  the  radius  of  the  earth  being   4000 
mi.,  that  of  Mars  2000  mi.,  and  the  mass  of  the  earth  nine  times  that 
of  Mars  ?     (The  accelerations  are  directly  proportional  to  the  masses 
and  inversely  proportional  to  the  squares  of  the  radii.) 

10.  If  a  body  weighs  45  Ib.  on  a  spring  balance  on  the  earth,  what 
would  it  weigh  on  Mars,  the  radius  of  the  earth  being  4000  mi.,  that 
of  Mars  2000  mi.,  and  the  mass  of  the  earth  nine  times  that  of  Mars  ? 

11.  How  far  above  the  earth's  surface  would  a  pound  ball  have  to 
be  taken  to  reduce  its  weight  to  1  oz.,  if  the  earth's  radius  is  4000 
mi.? 


V.     FALLING  BODIES 

123.  Rate  at  which  Different  Bodies  Fall.  —  It  is  a  familiar 
fact  that  heavy  bodies,  such  as  a  stone  or  a  piece  of  iron, 
fall  much  faster  than  such  light  bodies  as  feathers,  bits  of 
paper,  and  snow  crystals.  Before  the  time  of  Galileo  it 
was  supposed  that  different  bodies  fall  with  velocities  pro- 


FALLING  BODIES 


105 


portional  to  their  weights.  This  erroneous  notion  was 
corrected  by  Galileo  by  means  of  his  famous  experiment 
of  dropping  various  bodies  from  the  top  of  the  leaning 
tower  of  Pisa  (Fig.  103)  in  the  presence  of  professors  and 
students  of  the  uni- 
versity in  that  city. 
He  showed  that  bodies 
of  different  materials 
fell  from  the  top  of 
the  tower  to  the 
ground,  a  height  of 
180  feet,  in  practically 
the  same  time ;  also 
that  light  bodies,  such 
as  paper,  fell  with  ve- 
locities approaching 
more  and  more  nearly 
those  of  heavy  bodies 
the  more  compactly 
they  were  rolled  to- 
gether in  a  ball.  The 
slight  differences  in 
the  velocities  observed 
he  rightly  ascribed  to  the  resistance  of  the  air,  which  is 
relatively  greater  for  light  bodies  than  for  heavy  compact 
ones.  This  inference  Galileo  could  not  completely  verify 
because  the  air  pump  had  not  yet  been  invented. 

124.  Resistance  of  the  Air.  —  Place  a  small  coin  and  a  feather, 
or  a  shot  and  a  bit  of  tissue  paper,  in  a  glass  tube  from  4  to  6  feet 
long.  It  is  closed  at  one  end  and  fitted  with  a  stopcock  at  the  other 
(Fig.  104).  Hold  the  tube  in  a  vertical  position  and  suddenly  invert 
it;  the  coin  or  the  shot  will  fall  to  the  bottom  first.  Now  exhaust  the 
air  as  perfectly  as  possible ;  again  invert  the  tube  quickly ;  the  lighter 


Fig.  103 


106 


MECHANICS   OF  SOLIDS 


Fig.  104 


body  will  now  fall  as  fast  as  the  heavier  one.     This  experiment, 

known  as  the  "  Guinea  and  Feather  Tube,"  was  first  performed  by 
Newton.  It  demonstrates  that  if  the  resistance  of  the 
air  were  wholly  removed,  all  bodies  at  the  same  place 
would  fall  with  the  same  acceleration. 

An  interesting  modification  of  the  Newtonian  ex- 
periment is  the  following  :  Cut  a  round  piece  of  paper 
slightly  smaller  than  a  cent  and  drop  the  cent  and  the 
paper  side  by  side ;  the  cent  will  reach  the  floor  first. 
Then  lay  the  paper  on  the  cent  and  drop  them  in  that 
position ;  the  paper  will  now  fall  as  fast  as  the  cent. 
Explain. 

The  friction  of  the  air  against  the  surface  of  bodies 
moving  through 
it  limits  their  ve- 
locity. A  cloud 
floats,  not  because 
it  is  lighter  than 

the   atmosphere,  for   it  is  actu- 
ally   heavier,    but    because    the 

surface  friction    is    so    large  in 

comparison   with  the  weight  of 

the  minute  drops  of  water,  that 

the   limiting  velocity  of  fall   is 

very  small. 

When  a  stream  of  water  flows 

over  a  high  precipice,  it  is  broken 

into  fine  spray  and  falls  slowly. 

Such   is   the  explanation  of  the 

Staubbach  fall  (dust  brook)   at 

Lauterbrunnen    in     Switzerland 

(Fig.    105).      The    precipice    is 

300  m.  high,  and  the  fall  viewed 

from  its  face  resembles  a  magni- 
ficent transparent  veil,  kept  in 

movement  by  currents  of  air.  Fig-  105 

125.    Laws  of  Falling  Bodies.  —  Galileo  verified  the  fol- 
lowing laws  of  falling  bodies  :  — 


FALLING  BODIES  107 

I.  The  velocity  attained  by  a  falling  body  is  propor- 
tional to  the  time  of  falling. 

II.  The  distance  fallen  is  proportional  to  the  square  of 
the  time  of  descent. 

III.  The  acceleration  is  twice  the  distance  a  body  falls 
in  the  first  second. 

These  laws  will  be  recognized  as  identical  with  those 
derived  for  uniformly  accelerated  motion,  §§  91  and  92. 
If  the  inclined  plane  in  Galileo's  experiment  be  tilted  up 
steeper,  the  effect  will  be  to  increase  the  acceleration 
down  the  plane ;  and  if  the  board  be  raised  to  a  vertical 
position,  the  marble  will  fall  freely  under  gravity  and  the 
acceleration  will  become  g  (§  120). 

Since  the  acceleration  g  is  sensibly  constant  for  small 
distances  above  the  earth's  surface,  the  equations  already 
obtained  for  uniformly  accelerated  motion  may  be  applied 
directly  to  falling  bodies,  by  substituting  g  for  a  in  equa- 
tions (5)  and  (6).  Thus  we  have 

v  —  gt^     .     .     .     (Equation  11) 
and  s  =  |  fft2.      .     .     (Equation  12) 

If  in  equation  (12)  t  is  one  second,  s  =  ^  g ;  or  the  dis- 
tance a  body  falls  from  rest  in  the  first  second  is  half  the 
acceleration  of  gravity.  A  body  falls  490  cm.  or  16.08  ft. 
the  first  second ;  and  the  velocity  attained  is  980  cm.  or 
32.16  ft.  per  second. 

126.  Projection  Upward.  —  When  a  body  is  thrown  verti- 
cally upward,  the  acceleration  is  negative,  and  it  loses  each 
second  g  units  of  velocity  (980  cm.  or  32.16  ft.).  Hence, 
the  time  of  ascent  to  the  highest  point  is  the  time  taken 
to  bring  the  body  to  rest.  If  the  velocity  lost  is  g  units  a 
second,  the  time  required  to  lose  v  units  of  velocity  will  be 


108  MECHANICS   OF  SOLIDS 


the  quotient  of  v  by  #,  or 


time  of  ascent  =  veloci^  o 


acceleration  of  gravity 

In  symbols  t  =  -  -     •     •     •     (Equation  13) 

9 

For  example,  if  the  velocity  of  projection  upward  were 
1470  cm.  per  second,  the  time  of  ascent,  neglecting  the 
frictional  resistance  of  the  air,  would  be  -^g^,  or  1.5  sec- 
onds. This  is  the  same  as  the  time  of  descent  again  to  the 
starting  point ;  hence,  the  body  will  return  to  the  starting 
point  with  a  velocity  equal  to  the  velocity  of  projection  but 
in  the  opposite  direction.  In  this  discussion  of  projection 
upward,  the  resistance  of  the  air  is  neglected. 

Problems 

In  the  following  problems,  unless  otherwise  stated  in  the  problem, 
g  is  to  be  taken  as  980  cm.  or  32  ft.  per  second  per  second. 

1.  A  brick  falls  to  the  ground  from  the  top  of  a  chimney  64  ft. 
high.    What  will  be  the  time  of  descent,  and  with  what  velocity  will  it 
strike  the  ground?     (Use  equation  (12)  for  the  time,  and  equation 
(11)  for  the  velocity.) 

2.  A  stone  dropped  from  a  bridge  strikes  the  water  in  3  sec.  after 
leaving  the  hand.     Find  the  height  of  the  bridge  above  the  water. 

3.  If  a  stone  thrown  vertically  upward  returns  to  the  ground  in 
4  sec.,  how  high  does  it  ascend  ?       v 

4.  A  ball  is  thrown  over  a  tree  100  ft.  high.     How  long  before  it 
will  return  to  the  ground  ? 

5.  A  ball  fired  horizontally  reaches  the  ground  in  4  sec.     What 
was  the  height  of  the  point  from  which  it  was  fired  (§  114)  ? 

6.  A  cannon  ball  is  fired  horizontally  from  a  fort  at  an  elevation 
of  78.4  m.  above  the  level  of  the  neighboring  sea.     How  many  sec- 
onds before  it  will  strike  the  water  ? 

7.  An  iron  ball  was  dropped  from  an  aeroplane  moving  eastward 
at  the  rate  of  45  mi.  an  hour.     It  reached  the  ground  528  ft.  east  of 


CENTRIPETAL  AND   CENTRIFUGAL    FORCE        109 

the  vertical  line  through  the  point  at  which  it  was  dropped.     What 
was  the  elevation  of  the  aeroplane? 

8.  With  what  velocity  must  a  ball  be  fired  upward  to  rise  to  the 
height  of  the  Washington  Monument,  555  ft.,  and  how  long  will  it 
be  in  the  air  ? 

9.  An  inclined  plane  40  ft.  long  is  elevated  at  one  end  2  ft.     In 
what  time  will  a  ball  roll  down  it,  neglecting  all  resistance  ?     (The 
acceleration  down  the  plane  is  the  component  of  g  parallel  to  the  in- 
cline.) 

10.  A  body  slides  without  friction  down  an  inclined  plane  300  cm. 
long  and  24.5  cm.  high.  If  it  moves  40  cm.  during  the  first  second, 
what  is  the  computed  value  of  g  ? 

VI.  CENTRIPETAL  AND  CENTRIFUGAL  FORCE 

127.  Definition   of   Centripetal  and   Centrifugal  Force.  — 
Attach  a  ball  to  a  cord  and  whirl  it  aroundx  by  the  hand. 
The  ball  pulls  on  the  cord,  the  pull  increasing  with  the 
velocity  of  the  ball.     If  the  ball  is  replaced  by  a  heavier 
one,  with  the  same  velocity  the  -pull  is  greater.     If   a 
longer  cord  is  used,  the  pull  is  less  for  the  same  velocity 
in  the  circle. 

The  constant  pull  which  deflects  the  body  from  a  rectilinear 
path  and  compels  it  to  move  in  a  curvilinear  one  is  the  centrip- 
etal force.  The  resistance  which  a  body  offers,  on  account 
of  its  inertia,  to  deflection  from  a  straight  line  is  the  centrif- 
ugal force.  When  the  motion  is  uniform  and  circular, 
the  force  is  at  right  angles  to  the  path  of  the  body  around 
the  circle  and  constant. 

These  two  forces  are  the  two  aspects  of  the  stress  in 
the  cord  (third  law  of  motion),  the  action  of  the  hand  on 
the  ball,  and  the  reaction  of  the  ball  on  the  hand. 

128.  Value  of  Either  Force.  —  The  centripetal  accelera- 

a 

tion  for  uniform  circular  motion  (§  95)  is  a  =  — ,  where  v 


110  MECHANICS   OF  SOLIDS 

is  the  uniform  velocity  in  the  circle,  and  r  is  the  radius. 
Further,  in  §  114  the  relation  between  force  and  accelera- 
tion was  found  to  be,  force  equals  the  product  of  the  mass 
and  the  acceleration  imparted  to  it  by  the  force.  Hence 
we  have 

centripetal  force  =  mass  x  centripetal  acceleration, 

or  /=—  .    ^     ^     ^     (Equation  14) 

r 

This  relation  gives  the  value  of  either  the  centripetal 
or  the  equal  centrifugal  force  in  the  absolute  system  of 
measurement,  because  it  is  derived  from  the  laws  of  motion 
and  is  independent  of  gravity.  In  the  metric  system  m 
must  be  in  grams,  v  in  centimeters  per  second,  and  r  in 
centimeters  ;  /  is  then  in  dynes.  To  obtain  /  in  grams 
of  force,  divide  by  980  (§  100).  In  the  English  system, 
m  must  be  in  pounds,  v  in  feet  per  second,  and  r  in  feet  ; 
dividing  by  32.2,  the  result  will  be  in  pounds  of  force. 

For  example  :  If  a  mass  of  200  gm.  is  attached  to  a 
cord  1  m.  long  and  is  made  to  revolve  with  a  velocity 
of  140  cm.  per  second,  the  tension  in  the  cord  is 

200  ><  14°2  =  39,200  dynes  =  =  40  grams  of  force. 


Again  if  a  body  having  a  mass  of  10  Ib.  1  oz.  move  in 
a  circle  of  5  ft.  radius  with  a  velocity  of  20  ft.  per  second, 

then  the  centripetal  force  is  /=    /^  *        =25  pounds 

O  X  o-ij.^ 

of  force. 

129.  Illustrations  of  Centrifugal  Force.  —  Water  adhering  to 
the  surface  of  a  grindstone  leaves  the  stone  as  soon  as  the  centrif- 
ugal force,  increasing  with  the  velocity,  is  greater  than  the  adhesion 
of  the  water  to  the  stone.  Grindstones  and  flywheels  occasionally 
burst  when  run  at  too  high  a  speed,  the  latter  when  the  engine  runs 
away  after  a  heavy  load  is  suddenly  thrown  off.  When  the  centrip- 


THE  PENDULUM  111 

etal  force  is  insufficient  to  deflect  the  body  from  the  tangent  to  the 
circle,  the  body  flies  off  along  the  tangent  line.  A  stone  is  thrown  by 
whirling  it  in  a  sling  and  releasing  one  of  the  strings. 

A  carriage  or  an  automobile  rounding  a  curve  at  high  speed  is 
subject  to  strong  centrifugal  forces,  which  act  through  the  tires.  The 
centripetal  force  consists  solely  of  the  friction  between  the  tires  and 
the  ground.  If  the  friction  is  insufficient,  "  skidding  "  takes  place. 

Centrifugal  machines  are  used  in  sugar  refineries  to  separate  sugar 
crystals  from  the  syrup,  and  in  dyeworks  and  laundries  to  dry  yarn 
and  wet  clothes  by  whirling  them  rapidly  in  a  large  cylinder  with 
openings  in  the  side.  Honey  is  extracted  from  the  comb  in  a  similar 
way.  When  light  and  heavy  particles  are  whirled  together,  the  heav- 
ier ones  tend  toward  the  outside.  New  milk  is  an  emulsion  of  fat 
and  a  liquid,  and  the  fat  globules  are  lighter  than  the  liquid  of  the 
emulsion.  Hence,  when  fresh  milk  is  whirled  in  a  dairy  separator, 
the  cream  and  the  milk  form  dis- 
tinct layers  and  collect  in  separate 
chambers. 

When  a  spherical  vessel  containing 
some  mercury  and  water  is  rapidly 
whirled  on  its  axis  (Fig.  106),  both 
the  mercury  and  the  water  rise  and 
form  separate  bands  as  far  as  possible 

from  the  axis  of  rotation,  the  mercury 

,  .,  J  Fig.  106 

outside. 

The  figure  of  the  earth  is  an  oblate  spheroid,  flattened  at  the  poles. 
This  flattening  was  doubtless  caused  by  the  centrifugal  force  of  rota- 
tion when  the  earth  was  in  a  plastic  state,  before  it  reached  its  present 
more  rigid  condition.  Centrifugal  force  causes  the  water  of  the  ocean 
to  flow  toward  the  equatorial  regions,  exposing  lands  at  the  north 
which  would  be  covered  with  water  if  the  earth  were  stationary. 

VII.     THE  PENDULUM 

130.  Simple  Pendulum.  —  Any  body  suspended  so  as  to 
swing  about  a  horizontal  axis  is  a  pendulum.  A  simple 
pendulum  is  an  ideal  one.  It  may  be  defined  as  a  mate- 
rial particle  without  size  suspended  by  a  cord  without 
weight.  A  small  lead  ball  suspended  by  a  long  thread 


112 


MECHANICS  OF  SOLIDS 


without  sensible  mass  represents  very  nearly  a  simple 
pendulum.  When  at  rest  the  thread  hangs  vertically 
like  a  plumb  line;  but  if  the  ball  be  drawn  aside  and 
released,  it  will  oscillate  about  its  position  of  rest.  Its 
excursions  on  either  side  become  gradually  smaller ;  but 
if  the  arc  described  be  small,  the  period  of  its  swing  will 
remain  unchanged.  This  feature  of  pendular  motion  first 
attracted  the  attention  of  Galileo  while  watching  the  slow 
oscillations  of  a  "lamp"  or  bronze  chandelier,  suspended 
by  a  long  rope  from  the  roof  of  the  cathedral  in  Pisa. 
Galileo  noticed  the  even  time  of  the  oscillations  as  the 
path  of  the  swinging  chandelier  became  shorter  and 
shorter.  Such  a  motion,  which  repeats  itself  over  and 
over  in  equal  time  intervals,  is  said  to  be  periodic. 

131.  The  Motion  of  a  Pendulum.  —  AN  in  Fig.  107  is  a  nearly 
simple  pendulum  with  the  ball  at  N.  When  the  ball  is  drawn  aside 
to  the  position  13,  its  weight,  represented  by  BG, 
may  be  resolved  into  two  components,  BD  in  the 
direction  of  the  thread,  and  EC  at  right  angles 
to  it  and  tangent  to  the  arc  BNE.  The  latter 
is  the  force  which  produces  motion  of  the  ball 
toward  N. 

As  the  ball  moves  from  B  toward  N  the  com- 
ponent BC  becomes  smaller  and  smaller  and 
vanishes  at  N,  where  the  whole  weight  of  the 
ball  is  in  the  direction  of  the  thread.  In  falling 
from  B  to  JV,  the  ball  moves  in  the  arc  of  a  circle 
under  the  influence  of  a  force  which  is  greatest  at 
B  and  becomes  zero  at  N.  The  motion  is  there- 
fore accelerated  all  the  way  from  B  to  N,  but  not  uniformly.  The 
velocity  increases  continuously  from  B  to  N,  but  at  a  decreasing  rate. 
The  ball  passes  N  with  its  greatest  velocity  and  continues  on 
toward  E.  From  N  to  E  the  component  of  the  weight  along  the 
tangent,  which  is  always  directed  toward  N,  opposes  the  motion  and 
brings  the  pendulum  to  rest  at  E.  It  then  retraces  its  path  and  con- 
tinues to  oscillate  with  a  periodic  and  pendular  motion. 


Fig.  107 


THE  PENDULUM 


113 


132.  Definition  of  Terms.  —  The  center  of  suspension  is  the 
point  or  axis  about  which  the  pendulum  swings.     A  single 
vibration  is  the  motion  comprised  between 

two  successive  passages  of  the  pendulum 
through  the  lowest  point  of  its  path,  as 
the  motion  from  N  to  B  (Fig.  108)  and 
back  to  N  again.  A  complete  or  double 
vibration  is  the  motion  between  two  suc- 
cessive passages  of  the  pendulum  through 
the  same  point  and  going  in  the  same 
direction.  A  complete  vibration  is  double 
that  of  a  single  one.  The  period  of  vibra- 
tion is  the  time  consumed  in  making  a  Flg*  ! 
complete  or  double  vibration.  The  amplitude  is  the  arc 
SNor  the  angle  BAN. 

133.  Laws  of  the  Pendulum.  —  The   following   are   the 
laws  of  a  simple  pendulum  : 

I.  For  small  amplitudes,  the  period  of  vibration  is 
independent  of  the  amplitude. 

II.  The   period   of  vibration   is  proportional   to   the 
square  root  of  the  length. 

III.  The  period  of  vibration  is  inversely  proportional 
to  the  square  root  of  the  acceleration  of  gravity. 

One  of  the  earliest  and  most  important  discoveries  by 
Galileo  was  that  of  the  experimental  laws  of  the  motion 
of  a  pendulum,  made  when  he  was  about  twenty  years  of 
age.  This  was  long  before  their  theoretical  investigation. 

To  illustrate  law  I.  It  is  only  necessary  to  count  the  vibrations  of 
a  pendulum  which  take  place  in  some  convenient  time  with  different 
amplitudes.  Their  number  will  be  found  to  be  the  same.  This  re- 
sult will  hold  even  when  the  amplitudes  are  so  small  that  the  vibra- 
tions can  only  be  observed  with  a  telescope. 


114 


MECHANICS  OF  SOLIDS 


To  illustrate  law  II.     Mount  three  pendulums  (Fig.  109),  making 
the  lengths  1  m.,  \  m.,  and  \  m.  respectively.     Observe  the  period  of 
a  single   vibration  for   each.     They  will   be  1  sec., 
'F'p'r'T*  \  sec.,  and  \  sec.  nearly,  or  in  periods  proportional 

to  the  square  root  of  the  lengths. 

In  accordance  with  law  III  a  pendulum  oscillates 
more  slowly  on  the  top  of  a  high  mountain  than  at 
sea  level,  and  more  slowly  at  the  equator  than  at  the 
poles. 

134.  Center  of  Oscillation  —Insert  a  small 
staple  in  one  end  of  a  meter  stick,  and  suspend  it 
so  as  to  swing  as  a  pendulum  about  a  hori- 
zontal axis  through  the  staple  (Fig.  110). 
With  a  ball  and  a  thread  make  a  simple 
pendulum  that  will  vibrate  in  the  same 
period  as  the  meter  stick.  Measure  the 
length  of  this  pendulum  and  lay  it  off  on 
the  meter,  beginning  at  the  staple.  It 
will  extend  two  thirds  of  a  meter  down. 
Bore  a  hole  through  the  meter  stick  at 
the  point  thus  found,  and  suspend  it  as 

a  pendulum  by  means  of  a  pin   through    this  hole.     Its 

period  of  vibration  will  be  the  same  as  before. 

The  bar  is  a  compound  pendulum,  and  the  new  Fig  110 
axis  of  vibration  is  called  the  center  of  oscillation. 
The  distance  between  the  center  of  suspension  and  the 
center  of  oscillation  is  the  length  of  the  equivalent  simple 
pendulum  that  vibrates  in  the  same  period  as  the  com- 
pound pendulum.  The  centers  of  suspension  and  of  os- 
cillation are  interchangeable  without  change  of  period. 

135.  Center  of  Percussion.  —  Suspend  the  meter  bar  by  the 
staple  at  the  end  and  strike  it  with  a  soft  mallet  at  the  center  of 
oscillation.  It  will  be  set  swinging  smoothly  and  without  perceptible 
jar. 

Hold  a  thin  strip  of  wood  a  meter  long  and  four  or  five  centimeters 
wide  by  the  thumb  and  forefinger  near  one  end-  Strike  the  flat  side 


Fig.  109 


THE  PENDULUM 


115 


with  a  soft  mallet  at  different  points.  A  point  may  be  found  where 
the  blow  will  not  throw  the  wood  strip  into  shivers,  but  will  only  set 
it  swinging  like  a  pendulum. 

The  center  of  oscillation  is  also  called  the  center  of  per- 
cussion; if  the  suspended  body  be  struck  at  this  point  at 
right  angles  to  the  axis  of  suspension,  it 
will  be  set  swinging  without  jar.  A  base- 
ball club  or  a  cricket  bat  has  a  center  of 
percussion,  and  it  should  strike  the  ball  at 
this  point  to  avoid  breaking  the  bat  and 
"stinging"  the  hands. 

136.  Application  of  the  Pendulum.  —  Gali- 
leo's discovery  suggested  the  use  of  the 
pendulum  as  a  timekeeper.  In  the  com- 
mon clock  the  oscillations  of  the  pendu- 
lum regulate  the  motion  of  the  hands. 
The  wheels  are  kept  in  motion  by  a  weight 
or  a  spring,  and  the  regulation  is  effected 
by  means  of  the  escapement  (Fig.  111). 
The  pendulum  rod,  passing  between  the 
prongs  of  a  fork  a,  communicates  its  mo- 
tion to  an  axis  carrying  the  escapement, 
which  terminates  in  two  pallets  n  and  m. 
These  pallets  engage  alternately  with  the 
teeth  of  the  escapement  wheel  R,  one 
tooth  of  the  wheel  escaping  from  a  pallet 
every  double  vibration  of  the  pendulum, 
ment  wheel  is  a  part  of  the  train  of  the  clock ;  and  as 
the  pendulum  controls  the  escapement,  it  also  controls 
the  motion  of  the  hands. 


Fig.  1 1 1 
The  escape- 


137.    Seconds  Pendulum.  —  A  seconds   pendulum   is   one 
making  a  single  vibration  in  a  second.     Its  length  in  New 


116 


MECHANICS   OF  SOLIDS 


York  is  99.31  cm.  This  is  the  length  of  the  equivalent 
simple  pendulum  vibrating  seconds.  The  value  of  gravity 
g  increases  from  the  equator  to  the  poles,  and  the  length 
of  the  seconds  pendulum  increases  in  the  same  proportion. 

Questions  and  Problems 

1.  Why  can  a  boy  throw  a  stone  so  much  farther  with  a  sling  than 
without  it  ? 

2.  Why  can  an  athlete  throw  a  hammer  so  much  farther  than  he 
can  "  put  the  shot "  of  the  same  mass  V 

3.  Why  is  the  outer  rail  on  a  railway  curve  elevated  above  the 
inner  one  ? 

4.  Why  does  a  bicycle  rider  lean  inward  when  running  round  a 

curve  ? 

5.  A  string  attached  to  a  mass  of  100  gm.  broke 
when  the  mass  was  whirled  about  the  hand  at  a 
distance  of  1  m.,  and  at  the  rate  of  10  revolutions 
in    3    seconds.      Compute  the  breaking  force   in 
dynes. 

6.  A  mass  of  50  gm.  is  connected  to  a  fixed 
point  by  a  string  2  m.  long,  and  is  whirled  round 
in  a  circle  once  in  3  seconds.     Find  the  tension 
in  the  string  in  dynes ;  also  in  grams  of  force. 

7.  A  ball  was  mounted  to  swing  as  a  conical 
pendulum  (Fig.  112)  ;   its  mass  was  2  kgm.,  its 
distance  CA  from  the  center  of  its  circular  path 
was  50  cm.,  and  it  made  10  revolutions  in  5  seconds. 

What  horizontal  force  in  grams  would  be  necessary  to  hold  the  ball 
out  at  A  if  it  were  not  revolving  ? 

8.  Two  pendulums  of  the  same  length  have  different  bobs,  one 
of  lead,  and  the  other  of  aluminum.  Will  their  periods  be  the  same  ? 
Why? 


Fig.  112 


CHAPTER  VI 


MECHANICAL   WORK 
I.     WORK  AND  ENERGY 

138.  Work.  —  A  man  does  work  in  climbing  a  hill  by  lifting 
himself  against  the  pull  of  gravity ;  a  horse  does  work  in  drawing  a 
wagon  up  an  inclined  roadway ;  a  locomotive  does  work  in  hauling 
a  train  on  the  level  against  frictional  resistances ;  gravity  does  work 
against  the  inertia  of  the  mass  when 
it  causes  the  weight  of  a  pile  driver 
(Fig.  113)  to  descend  with  increasing 
velocity ;  steam  does  work  on  the 
piston  of  a  steam  engine  and  moves 
it  by  pressure  against  a  resistance ; 
the  electric  current  does  work  by 
means  of  a  motor  when  it  drives  an 
air  compressor  on  an  electric  car  and 
forces  air  into  a  compression  tank. 

Whenever  an  agent  exerts 
force  on  a  body  and  causes  the 
point  of  application  to  move  in 
the  direction  of  the  force,  the 
agent  is  said  to  do  mechanical 
work.  Unless  the  point  of  ap- 
plication of  the  force  has  a  com- 
ponent of  motion  in  the  direction  in  which  the  force  acts, 
no  work  in  a  physical  sense  is  done.  The  columns  in  a 
modern  steel  building  do  no  work,  though  they  sustain 
great  weight ;  the  pillars  supporting  a  pediment  over  a 
portico  do  no  work  ;  a  person  holding  a  weight  suffers 

117 


Fig.  113 


118  MECHANICAL   WORK 

fatigue,  but  does  no  work  in  the  sense  in  which  this  word 
is  used  in  physics,  where  it  is  employed  to  describe  the 
result  and  not  the  effort  made. 

Work  is  the  act  of  effecting  a  change  in  the  state  of  a  sys- 
tem against  a  resistance  which  opposes  the  change. 

139.  Measure  of   Mechanical  Work. — Mechanical  work 
is  measured  by  the  product  of  the  force  and  the  displace- 
ment of  its  point  of  application  in  the  direction  in  which 
the  force  acts,  or 

work  =  force  x  displacement. 
In  symbols  w  =/  x  *  .     .     .     .     (Equation  15) 

Since  force  is  equal  to  the  product  of  mass  and  accelera- 
tion (§  114), 

TF=  ma  x  s.    .     .     (Equation  16) 

140.  Units  of  Work.  —  Three  units  of  work  are  in  com- 
mon use  : 

1.  The  foot  pound,  or  the  work   done  by  a  pound  of 
force  working  through  a  distance  of  one  foot.     This  unit 
is  in  common  use  among  English  speaking  engineers.     It 
is  open  to  the  objection  that  it  is  variable,  since  a  pound 
of  force  varies  with  the  latitude  and  with  the  elevation. 

2.  The  kilogram  meter,  or  the  work  done  by  a  kilogram 
of  force  working  through  a  distance  of  one  meter.     It  is 
the  gravitational  unit  of  work  in  the  metric  system,  and 
varies  in  the  same  manner  as  the  foot  pound. 

3.  The   erg,1   or   the  work   done   by  a   dyne  working 
through  a  distance  of  one  centimeter.     The   erg   is  the 
absolute  unit  in  the  c.  g.  s.  system  and  is  invariable. 

Since  a  gram  of  force  is  equal  to  980  dynes  (§  100),  if 

1  The  erg  is  from  the  Greek  word  meaning  work. 


WORK  AND  ENERGY  119 

a  gram  mass  be  lifted  vertically  one  centimeter,  the  work 
done  against  gravity  is  980  erg^.  Hence  one  kilogram 
meter  is  equal  to  980  x  1000  x  100  =  98,000,000  ergs. 

The  mass  of  a  "nickel"  is  5  gm.  The  work  done  in  lifting  it 
through  a  vertical  distance  of  5  m.  is  the  continued  product  of  5,  500, 
and  980,  or  2,450,000  ergs.  The  erg  is  therefore  a  very  small  unit 
and  not  suitable  for  measuring  large  quantities  of  work.  For  such 
purposes  it  is  more  convenient  to  use  a  multiple  of  the  erg,  called  the 
joule.1  Its  value  is 

1  joule  =  107  ergjfz  10,000,000  ei 

Expressed  in  this  larger  unit,  the  work  done  in  lifting  the  "  nickel  " 
is  0.245  joule. 

141.  Power.  —  While  it  takes  time  to  do  work,  it  is 
plain  that  time  is  not  an  element  in  the  amount  of  work 
done.  To  illustrate  :  Suppose  a  ton  of  coal  is  lifted  by 
a  steam  engine  out  of  a  coal  mine  through  a  vertical  shaft 
300  ft.  deep.  The  work  is  done  by  means  of  a  wire  rope, 
which  the  engine  winds  on  a  drum.  If  now  the  drum  be 
replaced  by  another  of  twice  the  diameter,  and  running 
at  the  same  rate  of  rotation,  the  ton  of  coal  will  be  lifted 
in  half  the  time  ;  but  the  work  done  against  gravity  re- 
mains the  same,  namely,  600,000  ft.  Ib. 

In  an  important  sense  the  engine  as  an  agent  for  doing 
work  is  twice  as  effective  in  the  second  instance  as  in  the 
first.  Time  is  an  important  element  in  comparing  the 
capacities  of  agents  to  do  work.  Such  a  comparison  is 
made  by  measuring  the  power  of  an  agent. 

Power  is  the  time  rate  of  doing  work^  or 

work     f  x  8          ,^ 
power  ==  -7—  =  -  .   .     (Equation  17) 


The  unit  of  power  in  common  use  among  American  and 
1  From  the  noted  English  investigator  Joule. 


120  MECHANICAL    WORK 

English  engineers  is  the  horse  power  (H.  P.)  ;  it  is  the 
rate  of  working  equal  to  33,000  ft.  Ib.  per  minute,  or 
550  ft.  Ib.  per  second.  Hence 

H.  P.  =   {*'     .     .     (Equation  18) 
oou  x  c, 

in  which  /is  in  pounds  of  force,  s  in  feet,  and  t  in  seconds. 
In  the  c.g.s.  system,  the  unit  of  power  is  the  watt.1     It 
is  the  rate  of  working  equal  to  one  joule  per  second.     A 
kilowatt  (K.W.)  is  1000  watts.     Hence 


In  equation  (19)  /is  in  dynes,  s  in  centimeters,  and  t  in 
seconds. 

One  horse  power  equals  746  watts,  or  0.746  kilowatt 
(nearly  |  K.  W.).  To  convert  kilowatts  into  horse  powers 
approximately,  add  one  third  ;  to  convert  horse  powers 
into  kilowatts,  subtract  one  fourth.  For  example,  60  K.W. 
are  equal  to  80  H.P.,  and  100  H.P.  are  equal  to  75  K.W. 

The  power  capacity  of  dynamo  electric  generators  is 
now  universally  expressed  in  kilowatts  ;  the  steam  en- 
gines and  water  turbines  used  to  drive  these  generators, 
are  commonly  rated  in  the  same  unit  of  power  ;  so,  too, 
the  capacity  of  electric  motors  is  more  often  given  in 
kilowatts  than  in  horse  powers.  A  kilowatt  hour  means 
power  at  the  rate  of  a  kilowatt  expended  for  one  hour. 
Thus,  20  kilowatt-hours  mean  20  K.W.  for  one  hour,  or 
5  K.W.  for  four  hours,  etc. 

142.    Energy.  —  Experience  teaches  that  under  certain 
conditions  bodies  possess  the   capacity  for  doing  work. 
Thus,  a  body  of  water  at  a  high  level,  gas  under  pressure 
1  From  the  eminent  English  engineer,  James  Watt. 


WORK  AND  ENERGY  121 

in  a  tank,  steam  confined  in  a  steam  boiler,  and  the  air 
moving  as  a  wind,  are  all  able  to  do  work  by  means  of 
appropriate  motors.  In  general,  a  body  or  system  on 
which  work  has  been  done  acquires  increased  capacity 
for  doing  work.  It  is  then  said  to  possess  more  energy 
than  before.  Energy  is  the  capacity  for  doing  work.  It 
is  therefore  measured  in  the  same  units  as  work. 

143.  Potential  Energy.  — A  mass  of  compressed  air  in  an 
air  gun  tends  to  expand  ;    it  possesses  energy  and  may 
expend  it  in  propelling  a  bullet.     The  storage  of  ^nergy 
is  seen  also  in  the  lifted  weight  of  the  pile  drives,  the 
coiled-spring  of  the  clock,  the  bent  bow  of  the  archer, 
the  pent  up  waters  behind  a  dam,  the  chemical  changes 
in  a  charged  storage  battery,  and  the  mixed  charge  oK 
gasoline  vapor  and  air  in  the  cylinder  of  a  gas  engine. 

In  all  such  cases  of  the  storage  of  energy  a  stress  is 
present.  The  compressed  air  pushes  outward  in  the  air 
gun ;  gravity  tugs  at  the  lifted  weight ;  the  spring  tends 
to  uncoil  in  the  clock  ;  the  bent  bow  strives  to  unbend ; 
the  water  presses  against  the  dam  ;  the  electric  pressure 
is  ready  to  produce  a  current ;  and  the  explosive  gas 
mixture  awaits  only  a  spark  to  set  free  its  energy.  The 
energy  thus  stored  is  called  energy  of  stress,  or,  more  com- 
monly, potential  energy.  The  energy  of  an  elevated  body, 
of  bending,  twisting,  deformation,  of  chemical  separation, 
and  of  the  stress  in  a  magnetic  field  are  all  examples  of 
potential  energy. 

144.  Kinetic  Energy. — The  energy  which  a  body  has  in 
consequence   of   its   motion   is  known   as  kinetic  energy. 
The  descending  hammer  forces  the  nail  into  the  wood; 
the  rushing  torrent  carries  away  bridges  and  overturns 
buildings  ;    the  swift  cannon  ball,  by  virtue  of  its  high 


122  MECHANICAL    WORK 

speed,  demolishes  fortifications  or  pierces  the  harveyized 
steel  armor  of  a  battleship  ;  the  energy  stored  in  the 
massive  rotating  flywheel  keeps  the  engine  running  and 
does  work  after  the  steam  is  shut  off. 

Kinetic  energy  must  not  be  confused  with  force.  A 
mass  of  moving  matter  carries  with  it  kinetic  energy,  but 
it  exerts  no  force  until  it  encounters  resistance.  Energy 
is  then  transferred  to  the  opposing  body,  and  force  is  ex- 
erted only  during  the  transfer. 

145.  Measure  of  Energy. — Energy  is  measured  in  the 
same  terms  as  those  used  in  measuring  work.  In  general, 
potential  energy  is  the  measure  of  the  mechanical  work 
done  in  storing  the  energy,  or 

P.  E.  =/  x  *.     .     .     (Equation  20) 

The  potential  energy  of  a  body  of  water,  for  example,  of 
weight  w  and  at  an  elevation  A,  is  wh  gravitational  units. 

For  kinetic  energy,  suppose  a  force  /  to  act  on  a  mass 
m  for  a  period  of  time  t ;  the  measure  of  the  effect  is  the 
impulse  ft  (§  111).  If  the  velocity  acquired  from  rest  in 
the  time  t  is  v,  the  momentum  produced  is  mv.  By  the 
second  law  of  motion  impulse  equals  the  momentum  im- 
parted. Therefore 

ft  =  mv (a) 

A  constant  force  applied  to  a  body  gives  rise  to  uni- 
formly accelerated  motion ;  and  if  the  body  starts  from 
rest,  the  mean  velocity  is  J  v.  It  is  also  the  space  traversed 

divided  by  the  time,  or  -.     Hence,  equating  the  two  ex- 

t 

pressions  for  the  mean  velocity, 


WORK  AND  ENERGY  123 

Multiply  (a)   and   (6)   together,  member  by   member, 
and  the  result  is 

fs  =  ±mv*.   .     .     (Equation  21) 

But/s  measures  the  work  done  by  the  force  /on  the  mass 
m  to  give  to  it  the  velocity  v,  while  working  through  the 
distance  s;  and  since  the  kinetic  energy  acquired  by  a 
body  is  measured  by  the  work  done  on  it  to  give  it  motion, 
it,  follows  that  the  energy  of  the  mass  m  moving  with  the 
velocity  v  is  \  mv2,  or 

K.  E.  =  %mv*.     .     .     (Equation  22) 


If  m  is  expressed  in  grams  and  v  in  centimeters  per 
second,  the  result  is  in  ergs.  To  reduce  to  gram  centi- 
meters, divide  by  the  value  of  g  in  centimeters  per  second 
per  second,  or  980.  If  m  is  in  pounds  and  v  in  feet  per 
second,  to  obtain  the  energy  in  foot  pounds,  divide  by  the 
value  of  g  in  feet  per  second  per  second,  or  32.2. 

146.  Transformations  of  Energy.  —  When  a  bullet  is  shot 
vertically  upward,  it  gradually  loses  its  motion  and  its 
kinetic  energy,  but  gains  energy  of  position  or  potential 
energy.  When  it  reaches  the  highest  point  of  its  flight, 
its  energy  is  all  potential.  It  then  descends,  and  gains 
energy  of  motion  at  the  expense  of  energy  of  position. 
The  one  form  of  energy  is,  therefore,  convertible  into  the 
other. 

The  pendulum  illustrates  the  same  principle.  While 
the  bob  is  moving  from  the  lowest  point  of  its  path 
toward  either  extremity,  its  kinetic  energy  is  converted 
into  potential  energy  ;  the  reverse  transformation  sets  in 
when  the  pendulum  reverses  its  motion.  All  physical 
processes  involve  energy  changes,  and  such  changes  are 
in  ceaseless  progress. 


124  MECHANICAL   WORK 

147.  Conservation  of  Energy.  —  Whenever  a  body  gains 
energy  as  the  result  of  work  done  on  it,  it  is  always  at  the 
expense  of  energy  in  some  other  body  or  system.     The 
agent,  or  body,  which  does  work    always  loses   energy  ; 
the  body  which  has  work  done  on  it  gains  energy  equal  to 
the  work  done.     On  the  whole  there  is  neither  gain  nor 
loss  of  energy,  but  only  its  transfer  from  one  body  to  an- 
other.    Innumerable  facts  and  observations  show  that  it 
is  as  impossible  to  create  energy  as  it  is  to  create  matter. 
So  the  law  of  conservation  of  energy  means  that  no  energy 
is  created  and  none  destroyed  by  the  action   of  forces   we 
know  anything  about. 

148.  Dissipation  .of  Energy. — Potential   energy   is    the 
more  highly  available  or  useful  form  of  energy.     It  always 
tends  to  go  over  into  the  kinetic  type,  but  in  such  a  way 
that  only  a  portion  of  the  kinetic  energy  is  available  to 
effect  useful  changes  in  nature  or  in  the  mechanic  arts. 
The  remainder  is  dissipated  as  heat.      This  running  down 
of  energy  by  passing  into  an  unavailable  form  is  known  as 
the  dissipation  of  energy.     It  was  first  recognized  and  dis- 
tinctly stated  by  Lord  Kelvin  in  1859. 

The  capacity  which  a  body  possesses  for  doing  work 
does  not  depend  on  the  total  quantity  of  energy  which  it 
may  possess,  but  only  on  that  portion  which  is  available, 
or  is  capable  of  being  transferred  to  other  bodies.  In  the 
problems  of  physics  our  chief  concern  is  with  the  varia- 
tions of  energy  in  a  body  and  not  with  its  total  value. 

Questions  and  Problems 

1.  Why  will  a  cord  supporting  a  weight  generally  break  if  the 
weight  be  lifted  and  then  let  fall  ? 

2.  A  stick   resting  across  two  blocks  may  be  strong  enough  to 
bear  your  weight,  but  will  break  if  you  jump  on  it.     Explain. 


Lord  Kelvin  (Sir  William  Thomson),  1824-1907,  was  born 
at  Belfast.  He  graduated  at  Cambridge  in  1845  and  in  the  same 
year  received  the  appointment  of  professor  of  natural  philosophy 
in  the  University  of  Glasgow,  a  position  which  he  held  for  fifty- 
three  years.  He  was  one  of  the  greatest  mathematical  physicists 
of  his  day.  His  invention  of  the  astatic  mirror-galvanometer  and 
the  siphon-recorder  has  made  successful  marine  cables  a  reality. 
His  laboratory  for  the  use  of  students  was  the  first  of  the  kind  to 
be  established.  His  most  noteworthy  investigations  were  in  heat, 
energy,  and  electricity,  yet  there  is  scarcely  any  portion  of  physi- 
cal science  that  has  not  been  greatly  enriched  by  his  genius. 


QUESTIONS  AND  PROBLEMS  125 

3.  Lake  Tahoe,  in  the  Sierra  Nevada,  is  at  an  elevation  of  6225 
ft.  above  the  sea.     Account  for  the  energy  stored  there  in  the  water. 

4.  In  what  form  is  the  energy  for  driving  a  steamship  taken  on 
board?    Is  the  energy  driving  a  sailing  vessel  potential  or  kinetic  ?    In 
what  form  is  it  supplied  to  an  automobile  ;  to  an  aeroplane ;  to  a  man? 

5.  How  much  work  is  done  in  lifting  a  stone  weighing  100  kgm. 
to  the  top  of  a  building  20  m.  high?    What  is  its  potential  energy  at 
the  top  ? 

6.  A  rectangular  marble  slab  6  ft.  3  in.  long,  3  in.  thick,  and 
weighing  500  Ib.  lies  on  a  level  floor.     How  much  work  must  be  done 
to  set  it  vertically  on  end  ? 

7.  A  baseball  whose  mass  is  150  grn.  is  moving  with  a  velocity 
of  5000  cm.  per  second.     What  is  its  kinetic  energy  in  ergs  ?     How 
much  work  would  be  done  in  stopping  it? 

8.  What  is  the  kinetic  energy  of  a  5-gm.  bullet  when  fired  with 
a  velocity  of  300  m.  per  second? 

9.  A  force  of  200  dynes  moves  a  mass  of  100  gm.  through  a  dis- 
tance of  50  cm.  in  10  sec.     How  much  work  is  done  ? 

10.  A  pull  of  50  Ib.  moves  a  100-lb.   truck  through  a  distance  of 
200  ft.     How  much  work  in  foot  pounds  is  done  ? 

11.  A  load  weighing  2  tons  was  drawn  up  a  hill  half  a  mile  long 
by  a  traction  engine.     The  hill  was  100  ft.  high.     How  much  work 
was  done  against  gravity  ? 

12.  How  much  work  can  a  40  H.  P.  engine  do  in  an  hour  ?    How 
much  coal  can  it  lift  out  of  a  mine  400  ft.  deep  in  10  hrs.  ? 

13.  A  thousand-barrel  tank  at  a  mean  elevation  of  50  ft.  is  filled 
with   water.     How  much  work  was  done  in   filling  it,   assuming  a 
barrel  of  water  to  weigh  260  Ib.  ?    How  long  would  it  take  a  motor, 
working  at  a  2  H.  P.  rate,  to  pump  it  full  ? 

14.  An   electric  motor  rated  at  100  K.  W.   is  used  to  operate  a 
pump.      The    water  has  to  be  lifted  to    a  mean   height  of  100  m. 
How  many  liters  can  the  motor  pump  in  an  hour? 

15.  What  is  the  power  of  an  agent  that  lifts  1000  kgm.  10  m. 
high  in  10  min.  ?    Express  the  result  in  K.  W. 

16.  Express   in  joules   the  work  done  by  100  kgm.  of  force   in 
moving   a  mass  of  100  kgm.   through  a  distance  of  100  m.  in  the 
direction  of  the  force. 


126  MECHANICAL    WORK 

17.  The  average  pressure  of  steam  on  the  piston  of  a  steam  engine 
is  120  Ib.  of  force  per  square  inch.     The  area  of  the  piston  is  50  sq.  in. 
The  piston  travels  30  in.  during  one  complete  revolution  of  the  fly- 
wheel, and  the  flywheel  makes  220  revolutions  per  minute.     What  is 
the  H.P.  of  the  engine? 

18.  A  railway  train  weighs  250  tons,  and  the  resistance    to  its 
motion  on  a  level  track  is  15  Ib.  of  force  per  ton.     What  H.  P.  must 
the  locomotive  develop  to  maintain  a  speed  of  40  mi.  an  hour  on  the 
level? 

19.  How  many  H.  P.  are    transmitted  by  a  rope  passing  over  a 
wheel  33  ft.  in  circumference  and  making  one  revolution  per  second, 
the  tension  in  the  rope  being  100  Ib.  of  force  ? 

20.  A  ball  whose  mass  is  100  gm.  is  struck  with  a  club  and  is 
given  a  velocity  of  40  m.  per  second.     How  much  energy  is  imparted 
by  the  blow? 

21.  A  train  weighing  150  tons  and  running  at  the  rate  of  30  mi. 
an  hour  is  brought  to  rest  by  the  air  brakes  within  a  distance  of  500 
ft.     Find  the  force  of  the  brakes. 

22.  A  constant  resistance  of  1000  dynes  is  applied  to  a  body  of 
200  gm.  mass,  moving  with  a  velocity  of  6  m.  per  minute  and  brings 
it  to  rest.     How  far  did  the  body  move  after  the  resistance  was  ap- 
plied? 

II.    MACHINES 

149.  What  a  Machine  is.  —  A  machine  is  a  device  de- 
signed to  change  the  direction  or  the  magnitude  of  a 
force  required  to  do  useful  work,  or  one  to  transform  and 
transfer  energy. 

ILLUSTRATIONS.  —  By  the  use  of  a  single  pulley,  the  direction  of 
the  applied  force  may  be  changed,  so  as  to  lift  a  weight,  for  example, 
while  the  force  acts  in  any  desired  direction.  A  water  wheel  trans- 
forms the  potential  and  kinetic  energy  of  falling  water  into  mechan- 
ical energy  represented  by  the  energy  of  the  rotating  wheel.  A 
dynamo  electric  machine  transforms  mechanical  energy  into  the 
energy  of  an  electric  current,  and  an  electric  motor  at  a  distance 
transforms  the  electric  energy  back  again  into  mechanical  work. 


MACHINES  127 

150.  General  Law  of  Machines.  —  Every  machine  must 
conform  to  the  principle  of  the  conservation  of  energy ; 
that  is,  the  work  done  by  the  applied  force  equals  the  work 
done  in  overcoming  the  resistance,  except  that  some  of  the  ap- 
plied energy  may  be  dissipated  as  heat  or  may  not  appear 
in  mechanical  form.     A  machine  can  never  produce  an  in- 
crease of  energy  so  as  to  ^ive  out  more  than  it  receives. 

Denote  the  applied  force,  or  effort,  by  E  and  the  resist- 
ance by  M,  and  let  D  and  d  denote  the  distances  respectively 
through  which  they  work.  Then  from  the  law  of  conser- 
vation of  energy,  the  effort  multiplied  by  the  distance 
through  which  it  acts  is  equal  to  the  resistance  multiplied 
by  its  displacement,  or 

ED=Rd..     .     .     (Equation  23) 

151.  Friction.  —  Friction  is  the  resistance  which  opposes  an 
effort  to  slide  or  roll  one  body  over  another.     It  is  called  into 
action  whenever  a  force  is  applied  to  make  one  surface 
move  over  another.     Friction  arises  from  irregularities  in 
the  surfaces  in  contact  and  from  the  force  of  adhesion. 
It  is  diminished  by  polishing  and  by  the  use  of  lubricants. 

Experiments  show  that  friction  (a)  is  proportional  to 
the  pressure  between  the  surfaces  in  contact,  (5)  is  inde- 
pendent of  the  area  of  the  surfaces  in 
contact  within  certain  limits,  and  (<?) 
has  its  greatest  value  just  before  motion 
takes  place.     The  friction  of  a  solid 
rolling  on  a  smooth  surface  is  less  than 
when  it  slides.     Advantage  is  taken 
of  this  fact  to  reduce  the  friction  of 
bearings.     A  ball  bearing  (Fig.  114)  Fig'  114 

substitutes  the  rolling  friction  between  balls  and  rings  for 
the  sliding  friction  between  a  shaft  and  its  journal. 


128  MECHANICAL    WORK 

152.  Advantages  and  Disadvantages  of  Friction.  —  Friction 
has  innumerable  uses  in  preventing  motion  between  sur- 
faces in  contact.     Screws  and  nails  hold  entirely  by  fric- 
tion ;  we  are  able  to  walk  because  of  friction  between  the 
shoe  and  the  pavement ;  shoes  with  nails  in  the  heels  are 
dangerous  on  cast-iron  plates  because  the  friction  between 
smooth  iron  surfaces  is  small.     Friction  is  useful  in  the 
brake  to  stop  a  motor  car  or  railway  train,  in  holding 
the  driving  wheels  of  a  locomotive  to  the  rails,  and  in 
enabling  a  gasoline  engine  to  drive  an  automobile  by  fric- 
tion between  the  tires  and  the  street. 

On  the  other  hand  friction  is  also  a  resistance  opposing 
useful  motion,  and  whenever  motion  takes  place,  work 
must  be  done  against  this  frictional  resistance.  The 
energy  thus  consumed  is  converted  into  heat  and  is  no 
longer  available  for  useful  work. 

153.  Efficiency  of  Machines.  —  On  account  of  the  impos- 
sibility of  avoiding  friction,  every  machine  wastes  energy. 
The  work  done  is,  therefore,  partly  useful  and  partly  waste- 
ful.    The  efficiency  of  a  machine  is  the  ratio  of  the  useful 
work  done  by  it  to  the  total  work  done  by  the  acting  force, 
or 

useful  work  done 
efficiency 


total  energy  applied 

For  example,  an  effort  of  100  pounds  of  force  applied  to  a  machine 
produces  a  displacement  of  40  ft.  and  raises  a  weight  of  180  Ib.  20  ft. 
high.  Then  100  x  40  =  4000  ft.  Ib.  of  energy  are  put  into  the  ma- 
chine, and  the  work  done  is  180  x  20  =  3600  ft.  Ib. 


Hence  efficiency  =          =  0.9  =  90  per  cent. 

Ten  per  cent  of  the  energy  is  wasted  and  ninety  per  cent  recovered. 


MACHINES  129 

Since  every  machine  wastes  energy,  a  machine  which 
will  do  either  useful  or  useless  work  continuously  without 
a  supply  of  energy  from  without,  a  so-called  "  perpetual 
motion  machine,"  is  thus. clearly  impossible. 

Let  e  denote  the  efficiency  of  a  machine,  then  from  the 
relations  just  explained,  equation  (23)  becomes 

eED  =  ED.     .     .     .     (Equation  24) 

This  relation  is  the  strictly  correct  one  to  apply  to  all 
machines ;  but  in  most  problems  dealing  with  simple 
machines,  friction  is  neglected. 

154.  Simple  Machines.  —  All  machines  can  be  reduced  to 

six  mechanical  powers  or  simple  machines :  the  lever,  the 
pulley,  the  inclined  plane,  the  wheel  and  axle,  the  wedge, 
and  the  screw.  Since  the  wheel  and  axle  is  only  a  modi- 
fied lever,  and  the  wedge  and  the  screw  are  modifications 
of  the  inclined  plane,  the  mechanical  powers  may  be  re- 
duced to  three.  In  all  cases,  neglecting  friction,  the  law 
expressed  by  equation  (23)  holds  good. 

155.  Mechanical  Advantage.  —  A  man  working  a  pump 

handle  and  pumping  water  is  ^an  agent  applying  energy ;  • 
the  pump  and  the  water  compose  a  system  receiving  energy. 
In  a  simple  machine  the  force  exerted  by  the  agent  ap- 
plying energy,  and  the  opposing  force  <of  the  system  re- 
ceiving energy,  may  be  denoted  by  the~*twcTterms,  effort, 
E,  and  resistance,* R.  The  problem  in  simple  machines 
consists  in  finding  the  ratio  of  the  resistance  to  the 
effort. 

The  ratio  of  the  resisting  force  R  to  the  applied  force  E 
is  called  the  'mechanical  advantage  of  the  machine.  This 
ratio  may  always  be  expressed  in  terms  of  certain  parts  of 
simple  machines. 


130  MECHANICAL    WORK 

156.  Moment  of   a   Force.  —  In   the   application   of  the 
lever,  the  pulley,  or  the  wheel  and  axle  there  is  motion 
about  an  axis.     The  application  of  a   single   force   to   a 
body  with  a  fixed  axis  produces  rotation  only.     Examples 
are  a  door  swinging  on  its  hinges  and  the  flywheel  of  an 
engine. 

The  effect  of  a  force  in  producing  rotation  depends,  not 
only  on  the  value  of  the  force,  but  on  the  distance  of  its 
line  of  application  from  the  axis  of  rotation.  A  smaller 
force  is  required  to  close  a  door  when  it  is  applied  at  right 
angles  to  the  door  at  the  knob  than  when  it  is  applied 
near  the  hinge.  Also,  an  increase  in  the  speed  of  rota- 
tion of  a  flywheel  may  be  secured  either  by  increasing 
the  applied  force  or  by  lengthening  the  crank.  Both 
these  elements  of  effectiveness  are  included  in  what  is 
known  as  the  moment  of  a  force. 

The  moment  of  a  force  is  the  product  of  the  force  and  the 
perpendicular  distance  between  its  line  of  action  and  the  axis 

of  rotation.     Let  M  be   a  body  which 

may  rotate  about  an  axis  through  0 
(Fig.  115).  The  moment  of  the  force 
F  applied  at  B  in  the  direction  CB  is 
F  x  OB ;  applied  in  the  direction  AB, 
its  moment  is  F  x  OA.  The  point  0 
is  called  the  center  of  moments. 

A  moment  is  considered  positive  if  it 
produces  rotation  in  a  clockwise  direc- 
tion, and  negative  if  in  the  other.     If 
the  su^n  of  the  positive  moments  equals  that  of  the  negative 
moments,  there  is  equilibrium. 

157.  The  Lever.  —  The  lever  is  more  frequently  used 
than    any  other  simple  machine.     In    its   simplest  form 


M 


MACHINES 


131 


the  lever  is  a  rigid  bar  turning  about  a  fixed  axis  called 
the  fulcrum.  It  is  convenient  to  divide  levers  into 
three  classes,  distinguished 
by  the  relative  position  of 
the  fulcrum  with  respect 
to  the  two  forces.  In  the 
first  class  the  fulcrum  is  be- 
tween the  effort  H  and  the 
resistance  R  (Fig.  116)  ;  in 
the  second  class  the  resistance 
is  between  the  effort  and  the 
fulcrum ;  in  the  third  class 
the  effort  is  between  the  re- 
sistance and  the  fulcrum.  Fig.  116 

158.  Examples  of  Levers.  —  A  crowbar  used 'as  a  pry  (Fig. 
117)  is  a  lever  of  the  first  class,  but  when  used  to  lift  a  weight  with 
one  end  on  the  ground  (Fig.  118),  it  is  a  lever  of  the  second  class. 


Fig.  117 


p 

Fig.  118 


Scissors  are  double  levers  of  the  first  class.  So  also  are  the  tongs 
of  a  blacksmith,  and  those  used  in  chemical  laboratories  for  lift- 
ing crucibles  (Fig.  119).  The  forearm  when  it  supports  a .  weight 


Fig.  119 


Fig.  120 


in  the  extended  hand,  and  the  door  when  it  is  closed  by  pushing  it 
near  the  hinge,  are  examples  of  levers  of  the  third  class.  Nutcrackers 
(Fig.  120)  and  lemon  squeezers  are  double  levers  of  the  second  class. 


132 


MECHANICAL    WORK 


Fig.  121 


The  steelyard  (Fig.  121)  is  a  lever  of  the  first  class  with  unequal 
arms.     The  common  balance  (Fig.  122)  is  a  lever  of  the  first   class 
with  equal  arms.     The  two  weights  are^thus  also  equal. 
The  conditions  for  a  sensitive  balance,  to  show  a  small 
excess   of   weight  in   one  pan  over  that  in  the  other, 

are  small  friction  at  the 
"^""T""'""*°"'2f^:^      fulcrum,   a    light    beam, 
and   the   center  of  grav- 
ity only  slightly  lower  than  the  "  knife- 
edge  "  forming  the  fulcrum. 

159.  Mechanical  Advantage  of  the 
Lever. —In  Fig.  123  E  is  the 
effort,  R  the  resistance  or  weight 
lifted,  O  the  fulcrum,  and  AC  and  BO  the  lever  arms. 
Consider  the  lever  to 
be  weightless  and  to 
rotate  about  0  with- 
out friction ;  then  the 
moment  of  the  force 
E  about  the  fulcrum 
(§  156)  is  E  x  AC, 
and  that  of  the  force 
R  is  RxBC.  These 
two  forces  tend  to  pro- 
duce rotation  in  op- 
posite directions ;  for  equilibrium  their  moments  are 
therefore  equal,  that  is,  E  x  AQ—  Ex  BC;  from  which 

I  i  R  =AC 

E     BO 

i  i 

(Equation  25) 

Hence,    the    mechanical 
advantage    of   the    lever 
equals  the  inverse  ratio 
Fig.  123  of  its  arms. 


Fig.  122 


MACHINES 


133 


If  the  weight  of  the  lever  has  to   be  taken   into   ac- 
count, it  is  to  be  treated  as  a  force  acting  at  the  center 
of  gravity   of  the  lever,     A 
and  its  moment'  must  be      i1  •  ' 
added  to  that  of  the  force 
turning  the  lever  in  the 
same  direction  as  its  own 
weight. 


Fig.  124 


EXAMPLE.  —  The  weights 
Wl  and  W2  are  placed  at  dis- 
tances 5  and  8  units  respectively  from  0  (Fig.  124).  If  Wl  is  20 
lb.,  what  must  Wz  be  for  equilibrium?  By  the  principle  of  moments 
about  0, 


whence 


20  x  5  =  W2  x  8 


W2  =  12.5  lb. 


If  the  lever  is  uniform,  it  is  balanced  about  the  fulcrum  0  and 
its  moment  is  zero.  Suppose  the  weight  of  the  bar  to  be  1  lb.  and  its 
center  of  gravity  4  units  to  the  left  of  O.  The  equation  for  equi- 
librium would  then  be 


Whence 


20  x  5  +  1  x  4  =  Wa  x  8. 
W2  =  13  lb. 


160.   The  Wheel  and  Axle  consists  of  a  cylinder  and  a 
wheel  of  larger  diameter  usually  turning 
together  on  the  same  axis.     In  Fig.  125 
the  axle    passes   through    (7,  the  radius 
of   the  cylinder  is  BC,  and  that  of  the 
wheel  is   AC.     The   weights  P  and  W 
are  suspended  by  ropes  wrapped  around 
f  \f,          j-mur    ^e  circumference    of   the   two   wheels; 
I ')  iil        their   moments    about    the    axis    0   are 
Fig.  125  P  x  AC  and  W  x  BC  respectively.     For 


134 


MECHANICAL    WORK 


equilibrium  these  moments  are  equal,  that  is,  P  x  AC  = 
WxEC.     Hence, 


R  and  r  are  the  radii  of  the  wheel  and  the  axle  respec- 
tively.    The  weight  P  represents  the  effort  applied  at  the 


Fig.  126 


Fig.  127 


circumference  of  the  wheel,  and  the  weight  W  the  resist- 
ance at  the  circumference  of  the  axle.  Therefore,  the 
mechanical  advantage  of  the  wheel  and  axle  is  the  ratio  of 
the  radim  of  the  wheel  to  that  of  the  axle. 

161.   Applications.  —  The    old 

well  windlass  for  drawing  water 
from  deep  wells  (Fig.  126)  by  means 
of  a  rope  and  bucket  is  an  applica- 
tion of  the  principle  of  the  wheel 
and  axle.  In  the  windlass  a  crank 
takes  the  place  of  a  wheel  and  the 
length  of  the  crank  is  the  radius  of 
the  wheel. 

In  the  capstan  (Fig.  127)  the  axle 
is  vertical,  and  the  effort  is  applied 
by  means  of  handspikes  inserted  in 
holes  in  the  top. 

The  derrick  (Fig.  128)  is  a  form 
of  wheel  and  axle  much  used  for  raising  heavy  weights.  In  the  form 
shown  it  is  essentially  a  double  wheel  and  axle.  The  axle  of  the 


Fig.  128 


MACHINES 


135 


first  system  works  upon  the  wheel  of  the  second  by  means  of  the  spur 
gear.  The  mechanical  advantage  of  such  a  compound  machine  is  the  ratio 
of  the  product  of  the  radii  of  the  ivheels  to  the  product  of  the  radii  of 
the  axles. 

162.   The  Pulley  consists  of  a  wheel,  catted  a  sheave,  free 
to  turn  about  an  axle  in  a  frame,  called  a  block  (Fig.  129). 
The  effort  and  the  resistance  are  attached  to  a 
rope  which  moves  in  a  groove  cut  in  the  circum- 
ference of  the  wheel.     A  simple  fixed  pulley  is 
one   whose   axis   does  not  change  its  position; 
it  is  used  to  change  the  direction  of  the  applied 
force  (Fig.  130).     If  friction  and  the  rigidity 
of   the   rope   are   neglected, 
the   tension  in   the   rope   is 
everywhere   the   same ;    the 
effort  and  the  resistance  are 
then  equal  to  ~each  other  and 
the    mechanical   advantage   is   unity. 
In  the  movable  pulley  (Fig.  131) 
it   is    evident   that   the   weight  W  is 
supported  by  two 
parts  of  the  cord, 
one  half  of  it  by 


w 


Fig.  130 


means  of  the  hook  fixed  in  the 
beam  above  and  the  other  half  by  the 
effort  E  applied  at  the  free  end  of 
the  cord.  If  the  weight  is  lifted,  it 
rises  only  half  as  fast  as  the  cord 
travels. 

163.  Systems  of  Fixed  and  Movable 
Pulleys.  —  Fixed  and  movable  pulleys 
are  combined  in  a  great  variety  of 
ways.  The  most  common  is  the  one 


136 


MECHANICAL    WORK 


employing  a  continuous  cord.      Figure  132  represents  a 
combination  of  one  fixed  and  one  movable  pulley.     Figure 
133   illustrates  the  common  "block  and 
tackle,"  where  each  block  has  more  than 
one  sheave. 

164.  Mechanical  Advantage  of  the  Pulley. 
—  In  Fig.  134  the  cord  passes  in  suc- 
cession around  each  pulley.  It  is  evi- 
dent that  if  the  movable  pulley  and  the 
resistance  R  are  moved  toward  the  fixed 
pulley  a  distance  a,  each  cord  passing 
between  the  two  blocks  must  be  short- 
ened by  a  units.  The  effort  ^therefore 
travels  through  a  distance 
of  na  units,  n  being  the 

number  of  parts  to  the  cord  between  the 

two   pulleys.     Then  by  the  general   law 

of  machines  (§  150), 


Fig.  132 


whence 


X  na  =  R  x  a  ; 

^.  =  n.    .  (Equation  27) 


Hence,  when  a  continuous  cord  is  used,  the 
mechanical  advantage  of  the  pulley  is  equal 
to  the  number  of  times  the  cord  passes  to 
and  from  the  movable  block. 

It  should  be  noticed  that  n  is  equal 
to  the  entire  number  of  sheaves  in  the 
fixed  and  movable  blocks,  or  to  that  num- 
ber plus  one.  If  the  upper  block  in  Fig. 
134  were  the  movable  one,  that  is,  if  the 
system  were  inverted  so  that  the  effort  E 


Fig.  133 


MACHINES 


137 


is  upward,  n  would  be  equal  to  one  more  than  the  number 
of  sheaves. 

165.  The  Inclined  Plane.  —  If  a  body  rests  on  an  inclined 
plane  without  friction,  the  weight  of  the  body  acts  verti- 
cally downward,  while  the  reaction  of  the    « 

plane  is  perpendicular  to  its  surface ;  hence 
a  third  force  must  be  applied  to  maintain 
the  body  in  equilibrium  on  the  incline. 
This  force  may  be  applied  (1)  parallel  to 
the  face  of  the  plane ;  or  (2)  parallel  to 
the  base  of  the  plane. 

166.  Mechanical  Advantage  of  the  Inclined 
Plane.  —  Case  I:    When  the  effort  is  applied 
parallel  to  the  face  of  the  plane  (Fig.  135). 
The  most  convenient  way  to  find  the  relation 

between     the 

force    E  and 

the  weight  W 

of  the  body  D 

is  to  apply  the 

principle     of 

work  (§  139). 

Suppose  D  to 

be  moved  by  the  force  E  from  A  to  0.  Then  the  work 
done  by  E  is  E  x  AC.  Since  the  body  D  is  lifted  through 
a  vertical  distance  BO,  the  work  done  on  it  against 
gravity  is  W  x  BO.  Therefore,  E  x  AO  =  W  X  BO,  and 


W 
E 


AO 
BO 


(Equation  28) 


or,  the  mechanical  advantage,  when  the  effort  is  applied 
parallel  to  the  face  of  the  plane,  is  the  ratio  of  the  length 
of  the  plane  to  its  height. 


138 


MECHANICAL    WORK 


Case  II:    When  the  effort  is  applied  parallel  to  the  base 
of  the  plane  (Fig.  436).     In  expressing  the  work  done  by 

the  force  E  in  moving  the 
body  up  the  plane  from  A 
to  (7,  we  must  take  the  dis- 
placement measured  in  the 
direction  in  which  the  force 
acts.  This  displacement 
in  this  case  is  not  AO,  but 
Then  the  general  equation 


w 
Fig.  136 


AB,  the  base  of  the  plane. 


becomes  E  x  AB  =  W  x  BO,  and 

W =AB ^b 
E      BO     h 


s^  ,.          ^p., 

(Equation  29) 


Hence,  the  mechanical  advantage,  when  the  effort  is  applied 
parallel  to  the  base  qf  the  plane,  is  the  ratio  of  the  base  of 
the  plane  to  its  height. 

167.  The  Wedge  is   a  double  inclined   plane   with  the 
effort  applied  parallel  to  the  base  of  the  plane,  and  usu'ally 
by  a  blow  with  a  heavy  body  (Fig.  137).     Although  the 
principle  of  the  wedge  is  the  same  as  that  of  the  inclined 
plane,  yet  no  exact  state- 
ment   of    its    mechanical 

advantage  is  possible,  be- 
cause the  resistance  has 
no  definite  relation  to  the 
faces  of  the  planes,  and 
the  friction  cannot  be 
neglected.  Many  cutting  instruments,  such  as  the  ax 
and  the  chisel,  act  on  the  principle  of  the  wedge  ;  also 
nails,  pins,  and  needles. 

168.  The  Screw  is  a  cylinder,  on  the  outer  surface  of 
which  is  a  uniform  spiral  projection,  called  the  thread. 


MACHINES 


139 


The  faces  of  this  thread  are  inclined  planes.     If  a  long 

triangular  strip  of  paper   be  wrapped   around   a   pencil 

(Fig.    138),   with    the    base    of 

the  triangle  perpendicular  to  the 

axis   of   the    cylindrical    pencil, 

the  hypotenuse  of  the  triangle 

will  trace  a  spiral  like  the  thread 

of  a  screw. 

The  screw  (Fig.   139)  works 

in  a  block  called  a  nut,  on  the 

inner  surface  of  which  is  a  groove,  the  exact  counterpart 

of  the  thread.     The  effort  is  applied  at  the  end  of  a  lever 

or  wrench,  fitted  either  to  the  screw  or  to  the  nut.     When 

either  makes  a  complete  turn, 
the  screw  or  the  nut  moves 
through  a  distance  equal  to  that 
between  two  adjacent  threads, 
measured  parallel  to  the  axis  of 
the  screw  cylinder.  This  dis- 
tance, s  in  Fig.  140,  is  called 

the  pitch  of  the  screw.     It  is  usually  expressed  as  the 

number  of  threads  to  the  inch  or  to  the  centimeter. 


Fig.  139 


169.   Mechanical  Advantage  of  the  Screw.  —  Since  the  screw 
is  usually  combined  with  the 
lever,  the  simplest  method  of 
finding    the    mechanical    ad- 
vantage is  to  apply  the  prin- 
ciple  of    work,   as   expressed 
in    the    general   law   of   ma- 
chines (§  150).     If  the  pitch  Fig.  140 
be  denoted  by  s  and  the  resistance  overcome  by  R,  then, 
ignoring  friction,  the  work  done  against  R  in  one  revolution 


140 


MECHANICAL   WORK 


of  the  screw  is  R  x  s.  If  the  length  of  the  lever  is 
the  work  done  by  the  effort  Uin  one  revolution  is  E  x  2  j 
Whence  JE  x  2  irl  =  R  x  s,  or 


(Equation  30) 


Hence,  the  mechanical  advantage  of  the  screw  equals  the  ratio 
of  the  distance  traversed  by  the  effort  in  one  revolution  of 
the  screw  to  the  pitch  of  the  screw. 


Fig.  144 


Fig.  145 


170.    Applications  of  the  Screw.  —  The  jackscrew  (Fig.  141), 

the  letter  press  (Fig.  142),  the  vise  (Fig.  143),  and  the  screw  propeller 
of  a  ship  are  familiar  examples  of  the  use  of  the  screw. 

An  important  application  of  the  screw,  though  not  as  a  machine, 
is  that  for  measuring  small  dimensions.  The  wire  micrometer  (Fig. 
144)  and  the  spherometer  (Fig.  145)  are  instruments  for  this  purpose. 
In  both,  an  accurate  screw  has  a  head  divided  into  a  number  of  equal 


QUESTIONS  AND  PROBLEMS  141 

parts,  100  for  example,  so  as  to  register  any  portion  of  a  revolution. 
If  the  pitch  of  the  screw  is  1  mm.,  then  turning  the  head  through  one 
of  its  divisions  causes  the  screw  to  move  parallel  to  its  axis  0.01  mm. 

Questions  and  Problems 

1.  What  are  the  relative  positions  of  the  effort,  the  resistance,  and 
the  fulcrum  in  the  following  instruments  :  a  pump  handle,  a  pitch- 
fork, a  can  opener,  a  pair  of  sugar  tongs,  an  oar,  and  a  claw  hammer? 

2.  In  which  direction  does  friction  on  the  rails  act  on  the  wheels 
of  a  locomotive  ?    On  those  of  a  freight  car  ?    Does  it  act  in  the  same 
direction  on  the  front  and  the  rear  wheels  of  an  automobile  ? 

3.  Calculate  the  efficiency  of  a  machine  when  an  effort  of  50  Ib. 
of  force  acting  through  30  ft.  lifts  a  weight  of  200  Ib.  a  distance 
of  6  ft.  (§  153). 

4.  If  in  a  system  of  pulleys  a  tension  of  45  kgm.  of  force  is  applied 
to  the  rope  and  the  rope  is  drawn  60  ft.,  while  a  weight  of  250  kgm. 
is  lifted  10  ft.,  what  is  the  efficiency  of  the  system  (§  153)  ? 

5.  If  in  a  lever  of  the  first  class  a  weight  of  100  kgm.  is  placed  at 
a  distance  of  10  cm.  from  the  fulcrum,  what  weight  would  have  to  be 
placed  50  cm.  from  the  fulcrum  to  balance  it? 

6.  A  weight  of  50  Ib.  is  placed  15  in.  from  the  fulcrum  in  a  lever 
of  the  second  class.     The  effort  is  5  Ib.  of  force.    Find  the  length  of 
the  lever. 

7.  A  uniform  bar,  weighing  2  Ib.  to  the  foot,  is  20  ft.  long.     It  is 
used  as  a  lever  of  the  first  class  to  lift  a  weight  of  475  Ib.     The  ful- 
crum is  2  ft.  from  one  end.     Find  the  effort  necessary  to  balance  the 
weight  (§  156). 

8.  A  uniform  bar  2  m.  long  and  weighing  4  kgm.  has  weights  of 
7  kgm.  and  15  kgm.  suspended  at  its  two  ends.     Where  must  the  ful- 
crum be  placed  for  equilibrium  ? 

SUGGESTION.  Let  x  be  the  distance  of  the  fulcrum  from  the  weight  of  7 
kgm. ;  then  the  distance  of  the  center  of  gravity  of  the  bar  from  the  fulcrum 
is  x—  I,  and  that  of  the  weight  of  15  kgrn.  is  2  —  x. 

9.  In  a  wheel  and  axle  the  diameter  of  the  axle  is  40  cm.,  and  to 
it  is  attached  by  a  rope  a  weight  of  500  kgm.     The  axle  is  turned  by 
a  lever  1  m.  long.     Find  the  effort  necessary  for  equilibrium. 

10.  The  diameter  of  the  cylinder  of  a  ship's  capstan  is  12  in. 
What  force  would  have  to  be  applied  to  a  handspike  at  an  effective 


142  MECHANICAL    WORK 

distance  of  6  ft.  in  order  to  turn  the  capstan  and  lift  an  anchor 
weighing  2400  Ib.  ? 

11.  In  the  block  and  tackle  shown  in  Fig.  133  there  are  three 
sheaves  in  each  block.    What  weight  will  a  force  of  200  Ib.  lift, 
neglecting  friction  ? 

12.  How  many  sheaves  would  be  required,  in  a  system  like  that 
of  Fig.  133,  in  order  that  an  effort  of  100  Ib.  should  just  balance  a 
weight  of  800  Ib.? 

13.  A  cart  weighing  210  kgm.  is  to  be  pushed  up  an  inclined 
plane  by  a  force  of  15  kgm.     If  the  height  of  the  plane  is  5  m.,  what 
must  be  its  length,  neglecting  friction  ? 

14.  The  efficiency  of  an  inclined  plane  is  80  per  cent.     If  the 
length  of  the  plane  is  25  ft.  and  its  height  5  ft.,  what  effort  acting 
parallel  to  the  face  of  the  plane  will  be  required  to  move  a  body 
weighing  400  Ib.  up  the  plane  (§  153)  ? 

15.  The  screw  of  a  letter  press  has  five  threads  to  the  inch,  the 
diameter  of  the  wheel  is  12  in.,  and  the  effort  applied  to  it  is  40  Ib. 
of  force.     Neglecting  friction,  what  is  the  pressure  of  the  plate? 

16.  A  weight  of  1000  Ib.  is  raised  by  a  jackscrew.     What  force 
must  be  applied,  in  addition  to  the  force  required  to  overcome  friction, 
if  the  lever  is  2  ft.  long  and  the  screw  has  five  threads  to  the  inch  ? 

17.  The  radii  of  a  wheel  and  axle  are  5  ft.  and  5  in.  respectively. 
It  was  found  that  a  force  of  100  Ib.  can  lift  a  weight  of  960  Ib.     What 
weight  would  100  Ib.  of  force  lift  if  there  were  no  friction  ?    What  is 
the  efficiency  of  the  machine  ? 

18.  If  the  front  sprocket  wheel  of  a  bicycle  contains  24  sprockets 
and  the  rear  one  8,  how  far  will  one  complete  turn  of  the  pedals  drive 
a  28  in.  wheel  ? 

19.  The  diameter  of  the  large  driving  wheel  of  a  sewing  machine  is 
12.5  in.  and  that  of  the  small  driven  wheel  is  3  in.     If  the  slip  of  the 
belt  is  4  per  cent.,  how  many  stitches  does  the  machine  make  for 
every  up-and-down  movement  of  the  treadle  ? 

20.  An   automobile  engine   makes   900  revolutions  per    minute 
when  driving  the  shaft  direct.     The  spur  wheels  in  the  differential 
give  a  ratio  between  the  revolutions  of  the  shaft  and  that  of  the  axle 
of  one  to  four.     With  36  in.  wheels  the  slipping  on  the  ground  is 
enough  to   reduce  the  distance   traveled   every  revolution  to  9   ft. 
What  is  the  speed  of  the  automobile  in  miles  per  hour  ? 


CHAPTER  VII 


D'  D 


SOUND 
I.    WAVE  MOTION 

171.  Vibrations.  —  A  vibrating  or  oscillating  body  is  one 
which  repeats  its  limited  motion  at  regular  short  intervals 
of  time.     A  complete  or  double  vibration  is  the  motion  be- 
tween two  successive  passages  of  the  moving  body  through 
any  point  of  its  path  in  the  same  direction. 

If  we  suspend  a  ball  by  a  long  thread  and  set  it  swinging  like  a 
common  pendulum,  it  will  return  at  regular  intervals  to  the  starting 
point.  If  we  set  the  ball  moving  in  a 
circle,  the  string  will  describe  a  conical 
surface  and  the  ball  will  again  return  peri- 
odically to  the  point  of  departure. 

172.  Kinds    of   Vibration.  —  Clamp 
one  end   of  a  thin   steel  strip  in  a  vise 
(Fig.  146)  ;  draw  the  free  end  aside  and 
release  it.     It  will  move  repeatedly  from 
D'  to  D"  and  back  again.     The  shorter  or 
thicker  the  strip,  the  quicker  its  vibra- 
tions ;  when  it  becomes  like  the  prong  of 
a  tuning  fork,  it  emits  a  musical  sound. 

Vibrations  like  these  are  trans- 
verse.    A  body  vibrates  transversely  Fig.  146 
when  the  direction  of  the  motion  is 

at  right  angles  to  its  length.  The  strings  of  a  violin,  the 
reeds  of  a  cabinet  organ,  and  the  wires  of  a  piano  are 
familiar  examples. 

143 


144 


SOUND 


Fasten  the  ends  of  a  long  spiral  spring  securely  to  fixed  supports 
with  the  spring  slightly  stretched.     Crowd  together  a  few  turns  of 

the  spiral  at  one  end  and 
release  them.  A  vibratory 
movement  will  travel  from 


Fig.  147 


one  end  of  the  spiral  to  the 
other,    and    each   turn  of 

wire  will  swing  backward  and  forward  in  the  direction  of  the  length 

of  the  spiral  (Fig.  147). 

The  vibrations  of  the  spiral  are  longitudinal.  A  body 
vibrates  longitudinally  when  its  parts  move  backward  and 
forward  in  the  direction  of  its  length.  The  vibrations  set 
up  in  a  long  glass  tube  by  stroking  it  lengthwise  with  a 
damp  cloth  are  longitudinal ;  so  are  those  of  the  air  in  a 
trumpet  and  the  air  in  an  organ  pipe. 

173.  Wave  Motion.  —  Tie  one  end  of  a  soft  cotton  rope,  such 
as  a  clothesline,  to  a  fixed  support ;  grasp  the  other  end  and  stretch 
the  rope  horizontally.     Start  a  disturbance  by  an  up-and-down  motion 
of  the  hand.     Each  point  of  the  rope  will  vibrate  with  simple  har- 
monic motion  (§  97),  while  the  disturbance  will  travel  along  the  rope 
toward  the  fixed  end. 

This  progressive  form  or  change  in  shape,  due  to  the  peri- 
odic vibration  of  the  particles  of  the  medium  through  which 
it  moves,  is  a  wave.  The  particles  are  not  all  in  the  same 
phase  (§  175)  or  stage  of  vibration,  but  they  pass  through 
corresponding  positions  in  succession. 

174.  Transverse  Waves.  — A  small   camel's-hair  brush  is  at- 
tached to  the  end  of  a  long  slender  strip  of  clear  wood,  mounted  as 


Fig.  148 

shown  in  Fig.  148.     The  brush  should  just  touch  the  paper  under  it. 
Ink  the  brush  and  draw  the  movable  board  with  the  attached  paper 


WAVE  MOTION 


145 


under  the  brush  while  at  rest.  The  brush  will  mark  the  straight 
middle  line  running  through  the  curve  shown  in  the  figure.  Replace 
the  board  in  the  starting  position ;  then  pull  the  strip  aside  and 
release  it.  Again  draw  the  board  under  the  brush  with  uniform 
motion.  This  time  the  brush  traces  the  curved  line.  It  is  an  har- 
monic curve  or  graphic  wave  form. 

Suppose  a  series  of  particles,  originally  equidistant,  to 
vibrate  transversely  with  simple  harmonic  motion.  Let 
Fig.  149  represent  the  position  of  the  particles  at  some 


i 

1 

' 

C    T 

T 

M 

a       i     i 

i  M 

2 

1 

m  I  j  j  j 

i 
i 

j  i  i  ;  J 

*  '  ! 

i 
i 

|  !  i 

i 

i 

I  ' 

i  < 

Fig.  149 

particular  instant.  The  particle  at  g  has  reached  its 
maximum  displacement  in  one  direction  and  the  one  at  8 
the  maximum  in  the  other.  At  ra  the  particle  is  moving 
with  its  greatest  velocity  in  one  direction,  and  the  particle 
at  y  with  its  greatest  velocity  in  the  other  direction.  If 
the  wave  is  traveling  toward  the  right,  a  moment  later 
the  transverse  displacement  of  g  will  be  less  and  that  of  i 
a  maximum,  the  crest  of  the  wave  having  moved  forward 
from  g  to  i.  The  successive  particles  all  differ  in  phase 
by  the  same  amount. 

A  transverse  wave  is  one  in  which  the  vibration  of  the  par- 
ticles is  at  right  angles  to  the  direction  in  which  the  wave  is 
traveling. 


146 


SOUND 


175.    Longitudinal    Waves.  —  Place  a  lighted  candle  at  the 
conical  end  of  the  long  tin  tube  of  Fig.  150.     Over  the  other  end 


Fig.   150 

stretch  a  piece  of  parchment  paper.  Tap  the  paper  lightly  with  a 
cork  mallet ;  the  transmitted  impulse  will  cause  the  flame  to  duck, 
and  it  may  easily  be  blown  out  by  a  sharper  blow. 

The  air  in  the  tube  is  agitated  by  a  vibratory  motion, 
and  a  wave,  consisting  of  a  compression  followed  by  a 
rarefaction,  traverses  the  tube.  The  dipping  of  the  flame 
indicates  the  arrival  of  the  compression.  Each  particle 
of  air  vibrates  longitudinally  in  the  tube,  the  disturbance 
being  similar  to  that  of  the  vibrating  spiral. 


H 


E 


Fig.  151 

Figure  151  illustrates  the  distribution  of  the  air  particles 
when  disturbed  by  such  a  longitudinal  wave  of  com- 
pressions and  rarefactions.  B,  D,  F,  etc.,  are  regions  of 
compressions  ;  -A,  (7,  E,  etc.,  those  of  rarefaction.  The 
distances  of  the  different  points  of  the  curve  from  the 
straight  line  denote  the  relative  velocities  of  the  air 
particles.  A  and  (7,  or  B  and  D,  are  in  the  same  phase, 
that  is,  in  corresponding  positions  in  their  paths. 


WAVE  MOTION  147 

A  longitudinal  wave  is  one  in  which  the  oscillations  are 
backward  and  forward  in  the  same  direction  as  the  wave  is 
traveling. 

176.  Wave  Length.  —  The  length  of  a  wave  is  the  dis- 
tance   from    any   particle   to  the   next  one  in  the  same 
phase,  as  from  a  to  y  (Fig.  149),  or   from  A  to   O  or 
B  to  D  (Fig.  151).     Since  the  wave  form  travels  from  a 
to  y,  or  from  A  to   (7,   during  the  time  of  one  complete 
vibration  of  a  particle,  it  follows  that  the  wave  length 
is  the  distance  traversed  by  the  wave  during  one  vibration 
period. 

177.  Water  Waves.  —  One  of  the  most  familiar  examples  of 
transverse  waves  are  those  on  the  surface  of  water.     For  deep  water 
the  particles  describe  circles,  all  in  the  same  vertical  plane  containing 
the  direction  in  which  the  wave  is  traveling,  as  illustrated  in  Fig.  152. 


Fig.  152 

The  circles  in  the  diagram  are  divided  into  eight  equal  arcs,  and  the 
water  particles  are  supposed  to  describe  these  circles  in  the  direction 
of  watch  hands  and  all  at  the  same  rate  ;  but  in  any  two  consecutive 
circles  their  phase  of  motion  differs  by  one  eighth  of  a  period.  When 
a  has  completed  one  revolution,  b  is  one  eighth  of  a  revolution  behind 
it,  c  two  eighths  or  one  quarter,  etc.  A  smooth  curve  drawn  through 
the  positions  of  the  particles  in  the  several  circles  at  the  same  instant 
is  the  outline  or  contour  of  a  wave. 

When  a  particle  is  at  the  crest  of  a  wave,  it  is  moving  in  the  same 
direction  as  the  wave;  when  it  is  in  the  trough,  its  motion  is  opposite 
to  that  of  the  wave. 

The  crests  and  troughs  are  not  of  the  same  size,  and  the  larger  the 
circles  (or  amplitude),  the  smaller  are  the  crests  in  comparison  with 
the  troughs.  Hence  the  crests  of  high  waves  tend  to  become  sharp  or 
looped,  and  they  break  into  foam  or  white  caps. 


148  SOUND 


II.     SOUND  AND  ITS  TRANSMISSION 

178.  Sound  may  be  defined  as  that  form  of  vibratory  mo- 
tion in  an  elastic  medium  which  affects  the  auditory  nerves, 
and  produces  the  sensation  of  hearing.     All  the  external 
phenomena  of  sound  may  be  present  without  the  hear- 
ing ear.     Sound  should  therefore  be  distinguished  from 
hearing. 

179.  Source  of  Sound.  — If  we  suspend  a  small  elastic  ball  by  a 
thread  so  that  it  just  touches  the  edge  of  an  inverted  bell  jar,  and 
strike  the  edge  of  the  jar  with  a  felted  or  cork  mallet,  the  ball  will 
be  repeatedly  thrown  away  from  the  jar  as  long  as  the  sound  is  heard. 
This  shows  that  the  jar  is  vibrating  energetically. 

Stretch  a  piano  wire  over  the  table  and  a  little  above  it.  Draw 
a  violin  bow  across  the  wire,  and  then  touch  it  with  the  suspended 
ball  of  the  previous  paragraph.  So  long  as  the  wire  emits  sound 
the  ball  will  be  thrown  away  from  it  again  and  again. 

If  a  mounted  tuning  fork  (Fig.  153)  is  sounded, 
and  a  light  ball  of  pith  or  ivory,  suspended  by  a 
thread,  is  brought  in  contact  with  one  of  the  prongs 
at  the  back,  it  will  be  briskly  thrown  away  by  the 
energetic  vibrations  of  the  fork. 

Partly  fill  a  glass  goblet  with  water,  and  produce 
a  musical  note  by  drawing  a  bow  across  its  edge. 
The  tremors  of  the  glass  will  throw  the  surface  of 
the  water  into  violent  agitation  in  four  sectors,  with 
intermediate  regions  of  relative  repose.  This  agita- 
tion  disappears  when  the  sound  ceases. 

A  glass  tube,  four  or  five  feet  long,  may  be  made 
to  emit  a  musical  sound  by  grasping  it  by  the  middle  and  briskly 
rubbing  one  end  with  a  cloth  moistened  with  water.  The  vibrations 
are  longitudinal,  and  may  be  so  energetic  as  to  break  the  tube  into 
many  narrow  rings. 

Experiments  like  these  show  that  the  sources  of  sound 
are  bodies  in  a  state  of  vibration. 


SOUND  AND  ITS   TRANSMISSION  149 

180.  Media  for  Transmitting  Sound.  —Suspend  a  small  electric 
bell  in  a  bell  jar  on  the  air  pump  table  (Fig.  154).     When  the  air  is 
exhausted,  the  bell  is  nearly  inaudi- 
ble.    Sound  does  not  travel  through 

a  vacuum. 

Fasten  the  stem  of  a  tuning  fork  to 
the  middle  of  a  thin  disk  of  wood. 
Set  the  fork  vibrating,  and  hold  it 
with  the  disk  resting  on  the  surface 
of  water  in  a  tumbler,  standing  on 

a  table.     The  sound,  which  is  scarcely 

Fig.  154 
audible  when  there  is  no  disk  attached 

to  the  fork,  is  now  distinctly  heard  as  if  coming  from  the  table. 

Hold  one  end  of  a  long,  slender  wooden  rod  against  a  door,  and  rest 
the  stem  of  a  vibrating  fork  against  the  other  end.  The  sound  will  be 
greatly  intensified,  and  will  come  from  the  door  as  the  apparent  source. 

Press  down  on  a  table  a  handful  of  putty  or  dough,  and  insert  in  it 
the  stem  of  a  vibrating  fork  ;  the  vibrations  will  not  be  conveyed  to 
the  table  to  an  appreciable  extent. 

Only  elastic  media  transmit  sound,  and  some  transmit 
better  than  others. 

181.  Transmission  of  Sound  to  the  Ear. — Any  uninter- 
rupted series  of  elastic  bodies  will  transmit  sound' to  the 
ear,  be  they  solid,  liquid,  or  gaseous. 

A  bell  struck  under  water  sounds  painfully  loud  if  the  ear  of  the 
listener  is  also  under  water.  A  diver  under  water  can  hear  voices  in 
the  air.  By  placing  the  ear  against  the  steel  rail  of  a  railway,  two 
sounds  may  be  heard,  if  the  rail  is  struck  some  distance  away:  a 
louder  one  through  the  rails  and  then  another  through  the  air.  The 
faint  scratching  of  a  pin  on  the  end  of  a  long  stick  of  timber,  or  the 
ticking  of  a  watch  held  against  it,  may  be  heard  very  distinctly  if  the 
ear  is  applied  to  the  other  end. 

The  earth  conducts  sound  so  well  that  the  stepping  of  a  horse  may 
be  heard  by  applying  the  ear  to  the  ground.  This  is  understood  by 
the  Indians.  The  firing  of  a  cannon  at  least  200  miles  away  may  be 
heard  in  the  same  way.  The  report  of  a  mine  blast  reaches  a  listener 
sooner  through  the  earth  than  through  the  air. 


150  SOUND 

The  great  eruption  of  Krakatoa  in  1883  gave  rise  to  gigantic 
sound  waves,  which  produced  at  a  distance  of  2000  miles  a  report 
like  the  firing  of  heavy  guns. 

182.  Sound  Waves.  —  When   a  tuning  fork   or  similar 
body  is  set  vibrating,  the  disturbances  produced  in  the 
air  about  it  are  known  as  sound  waves.     They  consist  of 
a  series  of  condensations  and  rarefactions  succeeding  each 
other  at  regular  intervals.     Each  particle  of  air  vibrates 
in  a  short  path  in  the  direction  of  the  sound  transmission. 
Its  vibrations  are  longitudinal  as  distinguished  from  the 
transverse  vibrations  in  water  waves. 

183.  Motion  of  the  Particles  of  a  Wave.  —  The  motion  of 
the  particles  of  the  medium  conveying  sound  is  distinct 
from  the  motion  of  the  sound  wave.     A  sound  wave  is 
composed   of   a  condensation   followed   by  a  rarefaction. 
In  the  former  the  particles  have  a  forward  motion  in  the 
direction  in  which  the  sound  is  traveling  ;  in  the  latter 
they  have  a  backward   motion,  while   at   the  same  time 
both  condensation  and  rarefaction  travel  steadily  forward. 

The  independence  of  the  two  motions  is  aptly  illustrated  by  a 
field  of  grain  across  which  waves  excited  by  the  wind  are  coursing. 
Each  stalk  of  grain  is  securely  anchored  to  the  ground,  while  the 
wave  sweeps  onward.  The  heads  of  grain  in  front  of  the  crest  are 
rising,  while  all  those  behind  the  crest  and  extending  to  the  bottom 
of  the  trough  are  falling.  They  all  sweep  forward  and  backward, 
not  simultaneously,  but  in  succession,  while  the  wave  itself  travels  con- 
tinuously forward. 

III.    VELOCITY  OF  SOUND 

184.  Velocity  in  Air.  —  In  1822  a  scientific  commission  in 
France   made   experiments  to  ascertain   the   velocity   of 
sound   in   air.     Their   method    was   to   divide   into   two 
parties  at  stations  some  distance  apart,  and  to  determine 


VELOCITY  OF  SOUND  151 

the  interval  between  the  observed  flash  and  the  re- 
port of  a  cannon  fired  alternately  at  the  two  stations. 
The  mean  of  an  even  number  of  measurements  elimi- 
nates very  nearly  the  effect  of  the  wind.  The  final  result 
was  331  m.  per  second  at  0°  C.  The  defect  of  the 
method  is  that  the  perception  of  sound  and  of  light 
are  not  equally  quick,  and  they  vary  with  different 
persons. 

In  1871  Stone  at  Cape  Town  measured  the  velocity  of 
sound  by  stationing  two  observers  three  miles  apart  to 
give  signals  by  electricity  on  hearing  the  report  of  a  can- 
non. The  interval  between  these  signals  was  the  time 
required  for  the  sound  to  travel  the  intervening  three 
miles.  This  method  makes  use  of  the  sense  of  hearing 
only.  After  correcting  as  far  as  possible  for  all  sources 
of  error,  the  value  obtained  was  332.4  m.  or  1090.5  ft. 
per  second  at  0°  C.  Sound  travels  faster  at  a  higher 
temperature.  At  20°  C.  (68°  F.)  the  velocity  is  about 
1130  ft.  per  second.  The  correction  for  temperature  is 
0.6  m.,  or  nearly  2  ft.,  per  degree  C. 

185.  Velocity  in  Water.  —  In  1827  Colladon  and  Sturm, 
by  a  series  of  measurements  in  Lake  Geneva,  found  that 
sound  travels  in  water  at  the  rate  of  1435  m.  per 
second  at  a  mean  temperature  of  8.1°  C.  They  measured 
with  much  care  the  time  required  for  the  sound  of  a  bell 
struck  under  water  to  travel  through  the  lake  between 
two  boats  anchored  at  a  distance  apart  of  13,487  m.  It 
was  9.4  seconds. 

A  system  of  transmitting  signals  through  water  by  means  of  sub- 
merged bells  is  in  use  by  vessels  at  sea  and  for  offshore  stations. 
Special  telephone  receivers  have  been  devised  to  operate  under  water 
and  to  respond  to  these  sound  signals.  Indeed,  the  vessel  itself  acts 
as  a  sounding  board  and  as  a  very  good  receiver. 


152  SOUND 

186.  Velocity  in  Solids.  —  The  velocity  of  sound  in  solids 
is  in  general  greater  than  in  liquids  on  account  of  their 
high  elasticity  as  compared  with  their  density.  The  veloc- 
ity in  iron  is  5127  m.  per  second;  in  glass  5026  m. 
per  second  ;  but  in  lead  it  is  only  1228  m.  per  second,  at 
a  temperature  in  each  case  of  0°  C. 

Questions  and  Problems 

1.  Why  do  the  timers  in  a  200-yd.  dash  start  their  stop  watches 
by  the  flash  of  the  pistol  rather  than  by  the  report? 

2.  If  the  flash  of  a  gun  is  seen  3  sec.  before  the  report  is  heard, 
how  far  is  the  gun  from  the  observer,  the  temperature  being  20°  C.? 

3.  The  interval  between  seeing  a  flash  of  lightning  and  hearing 
the  thunder  was  5  sec. ;  the  temperature  was  25°  C.     How  far  away 
was  the  lightning  discharge  ? 

4.  Signals  given  by  a  gun  2  mi.  away  would  be  how  much  in 
error  when  the  temperature  is  20°  C.  and  the  wind  is  blowing  10  mi. 
an  hour  in  the  direction  from  the  listener  to  the  gun  ? 

5.  A  man  sets  his  watch  by  a  steam  whistle  which  blows  at  12 
o'clock.      The  whistle  is  1.5  mi.  away  and  the  temperature  15°  C. 
How  many  seconds  will  the  watch  be  in  error  ? 

6.  A  ball  fired  at  a  target  was  heard  to  strike  after  an  interval  of 
8  sec.     The  distance  of  the  target  was  1  mi.  and  the  temperature  of 
the  air  20°  C.     What  was  the  mean  velocity  of  the  ball  ? 

7.  The  distance  between  two  points  on  a  straight  stretch  of  rail- 
way is  2565  m.     An  observer  listens  at  one  of  these  points  and  a 
blow  is  struck  on  the  rails  at  the  other.     If  the  temperature  is  0°  C., 
what  is  the  interval   between   the  arrival  of  the  two  sounds,  one 
through  the  rails  and  the  other  through  the  air  ? 

8.  A  man  watching  for  the  report  of  a  signal  gun  saw  the  flash 
2  sec.  before  he  heard  the  report.     If  the  temperature  was  0°  C.  and 
the  distance  of  the  signal  gun  was  2225  ft.,  what  was  the  velocity  of 
the  wind? 

9.  A  shell  fired  at  a  target,  distance  half  a  mile,  was  heard  to 
strike  it  5  sec.  after  leaving  the  gun.     What  was  the  average  speed 
of  the  bullet,  the  temperature  of  the  air  being  20°  C.  ? 


REFLECTION  OF  SOUND  153 


IV.    REFLECTION  OF  SOUND 

187.  Echoes. — An  echo  is  the  repetition  of  a  sound  by 
reflection  from  some  distant  surface.     A  clear  echo  requires 
a  vertical  reflecting  surface,  the  dimensions  of  which  are 
large  compared  to  the  wave  length  of  the  sound.     A  cliff, 
a  wooded  hill,  or  the  broad  side  of  a  large  building  may 
serve  as  the  reflecting  surface.     Its  inequalities  must  be 
small  compared  to  the  length  of  the  incident  sound  waves  ; 
otherwise,  the  sound  is  diffused  in  all  directions.     A  loud 
sound  in  front  of  a  tall  cliff  an  eighth  of  a  mile  away  will 
be  returned  distinctly  after  about  a  second  and  a  sixth. 
If  the  reflecting  surface  is  nearer  than  abput  fifty  feet, 
the  reflected  sound  tends  to  strengthen  the  original  one, 
as  illustrated  by  the  greater  distinctness  of  sounds  indoors 
than  in  the  open  air.     In  large  rooms  where  the  echoes 
produce   a   confusion  of   sounds  the  trouble  may  be  di- 
minished by  adopting  some  method  to   prevent  regular 
reflection,  such  as  the  hanging  of  draperies. 

188.  Multiple  Echoes. — Parallel  reflecting  surfaces  at  a 
suitable  distance  produce  multiple  echoes,  as  parallel  mir- 
rors produce  multiple  images  (§  241).     The  circular  bap- 
tistry at  Pisa  and  its  spherical  dome  prolong  a  sound  for 
ten  or  more  seconds  by  successive  reflections ;    the  effect 
is  made  more  conspicuous  by  the  good  reflecting  surface 
of  polished  marble.     Extraordinary  echoes  sometimes  oc- 
cur between  the  parallel  walls  of  deep  canons. 

189.  Aerial  Echoes.  —  Whenever  the  medium  transmit- 
ting sound  changes  suddenly  in  density,  a  part  of  the 
energy  is  transmitted  and  a  part  reflected.     The  intensity 
of  the  reflected  system  is  the  greater  the  greater  the  dif- 
ference in  the  densities  of   the  two  media.     A  dry  sail 


154  SOUND 

reflects  a  part  of  the  sound  and  transmits  a  part ;  when 
wet  it  becomes  a  better  reflector  and  is  almost  impervious 
to  sound. 

Aerial  echoes  are  accounted  for  by  sudden  changes  of 
density  in  the  air.  Air,  almost  perfectly  transparent  to 
light,  may  be  very  opaque  to  sound.  When  for  any  rea- 
son the  atmosphere  becomes  unstable,  vertical  currents 
and  vertical  banks  of  air  of  different  densities  are  formed. 
The  sound  transmitted  by  one  bank  is  in  part  reflected 
by  the  next,  the  successive  reflections  giving  rise  to  a  cu- 
rious prolonging  of  a  short  sound.  Thus,  the  sound  of  a 
gun  or  of  a  whistle  is  then  heard  apparently  rolling  away 
to  a  great  distance  with  decreasing  loudness. 

190.  Whispering  Gallery.  —  Let  a  watch  be  hung  a  few  inches 
in  front  of  a  large  concave  reflector  (Fig.  155).  A  place  may  be 

found  for  the  ear  at  some  distance 
in  front,  as  at  E,  where  the  tick- 
ing of  the  watch  may  be  heard 
with  great  distinctness.  The 
sound  waves,  after  reflection  from 
the  concave  surface,  converge  to  a 
point  at  E. 

Pig   155  The  action  of  the  ear  trumpet 

depends  on  the  reflection  of  sound 

from  curved  surfaces ;  the  sides  of  the  bell-shaped  mouth  reflect  the 
sound  into  the  tube  which  conveys  it  to  the  ear. 

An  interesting  case  of  the  reflection  of  sound  occurs  in 
the  whispering  gallery,  where  a  faint  sound  produced  at 
one  point  of  a  very  large  room  is  distinctly  heard  at  some 
distant  point,  but  is  inaudible  at  points  between.  It  re- 
quires curved  walls  which  act  as  reflectors  to  concentrate 
the  waves  at  a  point.  Low  whispers  on  one  side  of  the 
dome  of  St.  Paul's  in  London  are  distinctly  audible  on 
the  opposite  side. 


RESONANCE  155 


V.    EESONANCE 

191.  Forced  Vibrations.  —  A  body   is   often   compelled 
to  surrender  its  natural  period  of  vibration,  and  to  vibrate 
with  more  or  less  accuracy  in  a  manner  imposed  on  it  by 
an  external  periodic  force.     Its  vibrations  are  then  said 
to  be  forced. 

Huyghens  discovered  that  two  clocks,  adjusted  to  slightly  different 
rates,  kept  time  together  when  they  stood  on  the  same  shelf.  The 
two  prongs  of  a  tuning  fork,  with  slightly  different  natural  periods 
on  account  of  unavoidable  differences,  mutually  compel  each  other 
to  adopt  a  common  frequency.  These  two  cases  are  examples  of 
mutual  control,  and  the  vibrations  of  both  members  of  each  pair  are 
forced. 

The  sounding  board  of  a  piano  and  the  membrane  of  a  banjo  are 
forced  into  vibration  by  the  strings  stretched  over  them.  The  top 
of  a  wooden  table  may  be  forced  into  vibration  by  pressing  against 
it  the  stem  of  a  vibrating  tuning  fork.  The  vibrations  of  the  table 
are  forced  and  it  will  respond  to  a  fork  of  any  period. 

192.  Sympathetic   Vibrations.  —  Place  two  mounted  tuning 
forks,  tuned  to  exact  unison,  near  each  other  on  a  table.     Keep  one 
of  them  in  vibration  for  a  few  seconds  and  then  stop  it ;  the  other 
one  will  be  heard  to  sound. 

In  the  case  of  these  forks,  the  pulses  in  the  air  reach 
the  second  fork  at  intervals  corresponding  to  its  natural 
vibration  period  and  the  effect  is  cumulative.  The  ex- 
periment illustrates  sympathetic  vibrations  in  bodies  hav- 
ing the  same  natural  period.  If  the  forks  differ  in  period, 
the  impulses  from  the  first  do  not  produce  cumulative 
effects  on  the  second,  and  it  will  fail  to  respond. 

Suspend  a  heavy  weight  by  a  rope  and  tie  to  it  a  thread.  The 
weight  may  be  set  swinging  by  pulling  gently  on  the  thread,  releas- 
ing it,  and  pulling  again  repeatedly  when  the  weight  is  moving  iu 
the  direction  of  the  pull. 


156 


SOUND 


Suspend  two  heavy  pendulums  on  knife-edges  on  the  same  stand, 
and  carefully  adjust  them  to  swing  in  the  same  period.  If  then 
one  is  set  swinging,  it  will  cause  the  other  one  to  swing,  and  will  give 
up  to  it  nearly  all  its  own  motion. 

When  the  wires'of  a  piano  are  released  by  pressing  the  loud  pedal, 
a  note  sung  near  it  will  be  echoed  by  the  wire  which  gives  a  tone  of 
the  same  pitch. 

A  number  of  years  ago  a  supension  bridge  of  Manchester  in  Eng- 
land was  destroyed  by  its  vibrations  reaching  an  amplitude  beyond 
the  limit  of  safety.  The  cause  was  the  regular  tread  of  troops  keep- 
ing time  with  what  proved  to  be  the  natural  rate  of  vibration  of  the 
bridge.  Since  then  the  custom  has  always  been  observed  of  break- 
ing step  when  bodies  of  troops  cross  a  bridge. 

193.  Resonance. — Resonance  is  the  reenf or  cement  of 
sound  by  the  union  of  direct  and  reflected  sound  waves. 

Hold  a  vibrating  tuning  fork 
over  the  mouth  of  a  cylindrical  jar 
(Fig.  156).  Change  the  length  of 
the  air  column  by  pouring  in  water 
slowly.  The  sound  will  increase 
in  loudness  until  a  certain  length 
is  reached,  after  which  it  becomes 
weaker.  A  fork  of  different  pitch 
will  require  a  different  length 
of  air  column  to  reenforce  its 
sound. 

The  "  sound  of  the  sea  "  heard 
when  a  sea  shell  is  held  to  the 
ear  is  a  case  of  resonance.  The 
mass  of  air  in  the  shell  has  a  vi- 
bration rate  of  its  own,  and  it 
amplifies  any  faint  sound  of  the 
same  period.  A  vase  with  a  long 
neck,  or  even  a  teacup,  will  also 
exhibit  resonance. 

The  box  on  which  a  tuning  fork  is  mounted  (Fig.  157)  is  a  reso- 
nator, designed  to  increa.se  the  volume  of  sound.  The  air  within  the 
body  of  a  violin  and  all  instruments  of  like  character  acts  as  a 


Fig.  156 


CHARACTERISTICS  OF  MUSICAL   SOUNDS          157 

resonator.  The  air  in  the  mouth,  the  larynx,  and  the  nasal  pas- 
sages is  a  resonator;  the  length  and  volume  of  this  body  of  air 
can  be  changed  at  pleasure  so  as  to  reenforce  sounds  of  different 
pitch. 

194.  The  Helmholtz  Resonator.  —  The  resonator  devised  by 
Helmholtz  is  spherical  in  form,  with  two  short  tubes  on  opposite 
sides  (Fig.  158).  The  larger  opening  A  is  the  mouth  of  the  resona- 


Fig.  157  Fig.  158 

tor;  the  smaller  one  B  fits  in  the  ear.  These  resonators  are  made  of 
thin  brass  or  of  glass,  and  their  pitch  is  determined  by  their  size. 
.When  one  of  them  is  held  to  the  ear,  it  strongly  reenforces  any  sound 
of  its  own  rate  of  vibration,  but  is  silent  to  others. 

VI.    CHARACTERISTICS  OF  MUSICAL  SOUNDS 

195.  Musical  Sounds. — Sounds  are  said  to  be  music'al 
when  they  are  pleasant  to  the  ear.  They  are  caused  by 
regular  periodic  vibrations.  A  noise  is  a  disagreeable 
sound,  either  because  the  vibrations  producing  it  are  not 
periodic,  or  because  it  is  a  mixture  of  discordant  elements, 
like  the  clapping  of  the  hands. 

Musical  sounds  have  three  distinguishing  characteris- 
tics: pitch,  loudness,  and  quality. 


158 


SOUND 


Fig.  159 


196.  Pitch.  — Mount  on  the  axle  of  a  whirling  machine 
(Fig.  159),  or  on  th'e  armature  of  a  small  electric  motor, 

a  cardboard  or  metal  disk  D  with  a  series 
of  equidistant  holes  in  a  circle  near  its 
edge.  While  the  disk  is  rotating  rapidly, 
blow  a  stream  of  air  through  a  small 
tube  against  the  circle  of  holes.  A  dis- 
tinct musical  tone  will  be  produced.  If 
the  experiment  be  repeated  with  the  disk 
rotating  more  slowly,  or  with  a  circle  of 
a  smaller  number  of  holes,  the  tone  will 
be  lower ;  if  the  disk  is  rotated  more 
rapidly,  the  tone  will  be  higher. 

The  air  passes  through  the  holes  in  a 
succession  of  puffs  producing  waves  in 
the  air.  These  waves  follow  one  another 
with  definite  rapidity,  giving  rise  to  the 
characteristic  of  sound  called  pitch.  We  conclude  that 
the  pitch  of  a  musical  sound  depends  only  upon  the  number 
of  pulses  which  reach  the  ear  per  second.  '  To  Galileo  be- 
longs the  credit  of  first  pointing  out  the  relation  of  pitch 
to  frequency  of  vibration.  He  illustrated  it  by  drawing 
the  edge  of  a  card  over  the  milled  edge  of  a  coin. 

197.  Relation  between  Pitch,  Wave  Length,  and  Velocity.  — 
I£  a  tuning  fork  makes  256  vibrations  per  second,  and 
in  that  time  a  sound  travels  in  air,  at  20°  C.,  a  distance 
of  344  m.,  then  the  first  wave  will  be  344  m.  from  the 
fork  when  it  completes  its  256th  vibration.     Hence,  in 
344  m.,  there  will  be  256  waves,  and  the  length  of  each 
will  be  HI  m.,  or  1.344  m.     In  general,  then, 

7         velocity 

wave  length,  =  -^ - — » 

frequency 


Hermann  von  Helmholtz  (1821-1894)  was  born  at  Potsdam. 
He  received  a  medical  education  at  Berlin  and  planned  to  be  a 
specialist  in  diseases  of  the  eye,  ear,  and  throat.  His  studies  soon 
revealed  to  him  the  need  of  a  knowledge  of  physics  and  mathe- 
matics. To  these  subjects  he  gave  his  earnest  attention  and  soon 
became  one  of  the  greatest  physicists  and  mathematicians  of  the 
nineteenth  century.  He  made  important  contributions  to  all  de- 
partments of  physical  science.  He  is  the  author  of  an  important 
work  on  acoustics  and  is  celebrated  for  his  discoveries  in  this 
field.  But  perhaps  his  most  useful  contribution  is  that  of  the 
ophthalmoscope,  an  instrument  of  inestimable  value  to  the  oculist 
in  examining  the  interior  of  the  eye. 


CHARACTERISTICS   OF  MUSICAL   SOUNDS          159 

i 

or  in  symbols,  I  =  -,    v  =  nl,  and  w=  -  .     .    (Equation  31) 


198.  Loudness.  —  The  loudness  of  a  sound  depends  on 
the  intensity  of  the  vibrations  transmitted  to   the   ear. 
The  energy  of  the  vibrations  is  proportional  to  the  square 
of  their  amplitude;  but  since  it  is  obviously  impracticable 
to  express  a  sensation  in  terms  of  a  mathematical  formula, 
it  is  sufficient  to  say  that  the  loudness  of  a  sound  increases 
with  the  amplitude  of  vibration. 

As  regards  distance,  geometrical  considerations  would 
go  to  show  that  the  energy  of  sound  waves  in  the  open 
decreases  as  the  square  of  the  distance  increases,  but  the 
actual  decrease  in  the  intensity  of  sound  is  even  greater 
than  this.  The  energy  of  sound  waves  is  gradually  dis- 
sipated by  conversion  into  heat  through  friction  and 
viscosity. 

199.  Quality.  —  Two  notes  of  the  same  pitch  and  loud- 
ness,  such  as  those  of  a  piano  and  a  violin,  are  yet  clearly 
distinguishable  by  the  ear.     This  distinction  is  expressed 
by  the  term  quality  or  timbre.     Helmholtz  demonstrated 
that  the  quality  of  a  note  is  determined  by  the  presence  of 
tones  of  higher  pitch,  whose  frequencies  are  simple  mul- 
tiples of  that  of  the  fundamental  or  lowest  tone.     These 
are  known  as  overtones. 

The  quality  of  sounds  differs  because  of  the  series  of 
overtones  present  in  each  case.  Voices  differ  for  this 
reason.  Violins  differ  in  sweetness  of  tone  because  the 
sounding  boards  of  some  bring  out  overtones  different 
from  those  of  others.  Even  the  untrained  ear  can  readily 
appreciate  differences  in  the  character  of  the  music  pro- 
duced by  a  flute  and  a  cornet.  Voice  culture  consists  in 
training  and  developing  the  vocal  organs  and  resonance 


160 


SOUND 


cavities,  to  the  end  that  purer  overtones  may  be  secured, 
and  greater  richness  may  by  this  means  be  imparted  to 
the  voice. 

VII.     INTERFERENCE  AND  BEATS 

200.  Interference.  — Hold  a  vibrating  tuning  fork  over  a  cylin- 
drical jar  adjusted  as  a 
resonator,  and  turn  the 
fork  on  its  axis  until  a 
position  of  minimum 
loudness  is  found.  In 
this  position  cover  one 
prong  with  a  pasteboard 
tube  without  touching 
(Fig.  160).  The  sound 
will  be  restored  to  nearly 
maximum  loudness,  be- 
cause the  paper  cylinder 
cuts  off  the  set  of  waves 
from  the  covered  prong. 

It   is  well   known 

that     the     loudness 
Fig;.  160  {.,_-,  -i       £ 

of    the    sound    of    a 

vibrating  fork  held  freely  in  the  hand  near  the  ear,  and 

turned    on    its    stem,    exhibits 

marked    variations.       In     four   \  /' 

\  / 

positions  the  sound  is  nearly  in-         \x  / 

audible.     Let  A,  B  (Fig.  161)  \  / 

be  the  ends  of  the  two  prongs.  — -W         <  ' 

¥%%          &%( 
They  vibrate  with  the  same  fre-  cm  Md 

J  i.i 

quency,  but  in  opposite   direc-  /'A        B  \ 

tions,  as  indicated  by  the  arrows.  / 

When   the    two   approach   each       /// 

other,  a  condensation  is  produced    / 

between  them,  and  at  the  same  Fig.  161 


\ 


INTERFERENCE  AND  BEATS 


161 


time  rarefactions  start  from  the  backs  at  c  and  d.  The 
condensations  and  rarefactions  meet  along  the  dotted  lines 
of  equilibrium,  where  partial  extinction  occurs,  because  a 
rarefaction  nearly  annuls  a  condensation.  When  the  fork 
is  held  over  the  resonance  jar  so  that  one  of  these  lines  of 
interference  runs  into  the  jar,  the  paper  cylinder  cuts  off 
one  set  of  waves,  and  leaves  the  other  to  be  reenforced  by 
the  air  in  the  jar. 

Interference  is  the  superposition  of  two  similar  sets  of 
waves  traversing  the  medium  at  the  same  time.  One  of  the 
two  sets  of  similar  waves  may  be  direct  and  the  other  re- 
flected. If  two  sets  of  sound  waves  of  equal  length  and 
amplitude  meet  in  opposite  phases,  the  condensation  of  one 
corresponding  with  the  rarefaction  of  the  other,  the  sound 
at  the  place  of  meeting  is  extinguished  by  interference. 

201.  Beats.  —  Place  near  each  other  two  large  tuning  forks  of 
the  same  pitch  and  mounted  on 
resonance  boxes.  When  both  are 
set  vibrating,  the  sound  is  smooth, 
as  if  only  one  fork  were  sounding. 
Stick  a  small  piece  of  wax  to  a 
prong  of  one  fork;  this  load  in- 
creases its  periodic  time  of  vibra- 
tion, and  the  sound  given  by  the 
two  is  now  pulsating  or  throbbing. 

Mount  two  organ  pipes  of  the 
same  pitch  on  a  bellows,  and  sound 
them  together.  If  they  are  open 
pipes,  a  card  gradually  slipped  over 
the  open  end  of  one  of  them  will 
change  its  pitch  enough  to  bring 
out  strong  pulsations. 

With  glass  tubes  and  jet  tubes 
set  up  the  apparatus  of   Fig.  162. 
One  tube   is  fitted  with  a  paper  slider  so  that  its  length    may  be 
varied.     When  the  gas  flame  is  turned  down  to  the  proper  size,  the 


162  SOUND 

tube  gives  a  continuous  sound  known  as  a  "  singing  flame."  By 
making  the  tubes  the  same  length,  they  may  be  made  to  yield  the 
same  note,  the  combined  sound  being  smooth  and  steady.  Now 
change  the  position  of  the  slider,  and  the  sound  will  throb  and  pul- 
sate in  a  disagreeable  manner. 

These  experiments  illustrate  the  interference  of  two 
sets  of  sound  waves  of  slightly  different  period.  The  out- 
bursts of  sound,  followed  by  short  intervals  of  comparative 
silence,  are  called  beats. 

202.  Number  of  Beats.  —  If  two  sounds  are  produced  by 
forks,  for  example,  making  100  and  110  vibrations  per 
second  respectively,  then  in  each  second  the  latter  fork 
gains  ten  vibrations  on  the  former.     There  must  be  ten 
times  during  each  second  when  they  are  vibrating  in  the 
same  phase,  and  ten  times  in  opposite  phase.     Hence,  in- 
terference of  sound  must  occur  ten  times  a  second,  and 
ten  beats  are  produced.     Therefore,  the  number  of  beats 
per  second  is  equal  to  the  difference  of  the  vibration  rates 
(frequencies)  of  the  two  sounds. 

VIII.     MUSICAL  SCALES 

203.  Musical  Intervals.  —  A  musical  interval  is  the  rela- 
tion between  two  notes  expressed  as  the  ratio  of  their  fre- 
quencies of   vibration.      Many   of   these    intervals   have 
names  in  music.     When  the  ratio  is  1,  the  interval  is 
called  unison;    2,  an  octave;  f,  a  fifth;  |,  a  fourth;  etc. 
Any  three  notes  whose  frequencies  are  as  4:5:6  form  a 
major  triad,  and  alone  or  together  with  the  octave  of  the 
lowest  note,  a  major  chord.     Any  three  notes  whose  fre- 
quencies are  as  10  : 12  :  15  form  a  minor  triad,  and  alone  or 
with  the  octave  of  the  lowest,  a  minor  chord. 

Mount  the  disk  of  Fig.  163  on  the  whirling  table  of  Fig.  159. 
The  disk  is  perforated  with  four  circles  of  equidistant  holes,  number- 


MUSICAL   SCALES 


163 


Fig.   163 


ing  24,  30,  36,  and  48  respectively.     These  are  in  the  relation  of  4, 

5,  6,  8.     Rotate  with  uniform  speed,  and  beginning  with  the  inner 

circle,  blow  a  stream  of  air  against  each 

row  of  holes  in  succession.     The  tones 

produced  will  be  recognized  as  do,  mi, 

sol,  do',  forming  a  major  chord.     If  now 

the  speed  of  rotation  be  increased,  each 

note  will  rise  in  pitch,  but  the  musical 

sequence  will  remain  the  same. 

It  will  be  seen  from  the  fore- 
going relations  that  harmonious 
musical  intervals  consist  of  very 
simple  vibration  ratios. 

204.  The  Major  Diatonic  Scale. — 

A  musical  scale  is  a  succession  of  notes  by  which  musical 
composition  ascends  from  one  note,  called  the  keynote,  to 
its  octave.  This  last  note  in  one  scale  is  regarded  as  the 
keynote  of  another  series  of  eight  notes  with  the  same 
succession  of  intervals.  In  this  way  the  series  is  extended 
untilthe  limit  of  pitch  established  in  music  is  reached. 

The  common  succession  of  eight  notes,  called  the  major 
diatonic  scale,  was  adopted  about  three  hundred  and  fifty 
years  ago.  The  octave  beginning  with  middle  0  is 
written 

cf     d'     e'    f     g'     a1     V     c" 

The  three  major  triads  for  the  keynote  of  C  are  : 

c'  :   e'  :  g'  ] 

g1  :    bf  :    d"      ::4:  5:  6 
/'   :   a'  :    c"  \ 

The  frequency  universally  assigned  to  c'  in  physics  is 
256.  It  is  convenient  because  it  is  a  power  of  2,  and  it 
is  practically  that  of  the  "middle  (7"  of  the  piano.  If  cf 
is  due  to  256,  or  m,  vibrations  per  second,  the  frequency 


164  SOUND 

of  the  other  notes  of  the  diatonic  scale  may  be  found  by 
proportion  from  the  three  triads  above  ;  they  are  as 
follows  : 

256  288  320  3411  334  426f  480  512 

c1  d'  e'          f  g'          a1  V  c" 

do  re  mi  fa  sol         la  si  do 

m  -I  m  4m  4  m  tw  4m  -V-m  2m 

8  *t  O  X  5  O 

If  the  fractions  representing  the  relative  frequencies  be 
reduced  to  a  common  denominator,  the  numerators  may 
be  taken  to  denote  the  relative  frequencies  of  the  eight 
notes  of  the  scale.  They  are 

24     27     30^32     36     40     45     48 

An  examination  of  these  numbers  will  show  that  there 
are  only  three  intervals  from  any  note  to  the  next  higher. 
They  are  -| ,  a  major  tone ;  -i^0-,  a  minor  tone  ;  and  if,  a 
half  tone.  The  order  is  f ,  -^o,  if,  f ,  -L0-,  f,  if. 

205.  The  Tempered  Scale.  —  If  C  were  always  the  key- 
note, the  diatonic  scale  would  be  sufficient  for  all  purposes 
except  for  minor  chords  ;  but  if  some  other  note  be  chosen 
for  the  keynote,  in  order  to  maintain  the  samev order  of 
intervals,  new  and  intermediate  notes  will  have  10  be  in- 
troduced. For  example,  let  D  be  chosen  for  the  key- 
note, then  the  next  note  will  be  288  x  f  =  324  vibrations, 
a  number  differing  slightly  from  E.  Again,  324  x  -\^ 
=  39CL  a  note  differing  widely  from  any  note  in  the  series. 
In  like  manner,  if  other  notes  are  taken  as  keynotes,  and 
a  scale  is  built  up  with  the  order  of  intervals  of  the  dia- 
tonic scale,  many  more  new  notes  will  be  needed.  This 
interpolation  of  notes  for  both  the  major  and  minor  scales 
would  increase  the  number  in  the  octave  to  seventy-two. 


MUSICAL   SCALES 


165 


In  instruments  with  fixed  keys  such  a  number  is  un- 
manageable, and  it  becomes  necessary  to  reduce  the 
number  by  changing  the  value  of  the  intervals.  Such  a 
modification  of  the  notes  is  called  tempering.  Of  the  sev- 
eral methods  proposed  by  musicians,  that  of  equal  tempera- 
ment is  the  one  generally  adopted.  It  makes  all  the 
intervals  from  note  to  note  equal,  interpolates  one  note  in 
each  whole  tone  of  the  diatonic  scale,  and  thus  reduces 
the  number  of  intervals  in  the  octave  to  twelve.  The 
only  accurately  tuned  interval  in  this  scale  is  the  octave ; 
all  the  others  are  more  or  less  modified.  The  following 
table  shows  the  differences  between  the  diatonic  and  the 
equally  tempered  scales : 


Diatonic 
Tempered 


c' 
.  256 
.  256 

rf)  

d'         e'           f         g>          a' 
288       320        341.3     384       426 
287.3     322.5     341.7     383.6     430 

hfcr 

iTD            9 

Ifto 

1            <?            ^ 

^ 

O          Y           i            ]            ! 

Q                        *?                              1 

I            I                      1 
1            1           1           1           1 

c'  \  d'\    c' 

m  j 

Fig.   lo4X 

b' 

480 
483.3 


c" 
512 
512 


Figure  1 64  illustrates  the  scale  of  C  on  the  staff  and  the 
keyboard. 

206.  Limits  of  Pitch.  —  The  international  pitch,  now  in 
general  use  in  Europe  and  America,  assigns  to  a1  the  vi- 
bration frequency  of  435.  In  the  modern  piano  of  seven 
octaves  the  bass  A  has  a  frequency  of  about  27.5,  the 
highest  A,  3480. 

The  gravest  note  of  the  organ  is  the  C  of  16  vibrations 


166  SOUND 

per  second ;  the  highest  note  is  the  same  as  the  highest 
note  of  the  piano,  the  third  octave  above  a',  with  a  fre- 
quency of  3480. 

The  limits  of  hearing  far  exceed  those  of  music.  The 
range  of  audible  sounds  is  about  eleven  octaves,  or  from 
the  Q  of  16  vibrations  to  that  of  32,768,  though  many 
persons  of  good  hearing  perceive  nothing  above  a  fre- 
quency of  16,384,  an  octave  lower. 

Questions  and  Problems 

1.  Why  is  the  pitch  of  the  sounds  given  by  a  phonograph  raised 
by  increasing  the  speed  of  the  cylinder  or  the  disk  containing   the 
record  ? 

2.  A  megaphone  or  a  speaking  tube  makes  a  sound  louder  at  a 
distance.     Explain  why. 

3.  The  teeth  of  a  circular  saw  give  a  note  of  high  pitch  when 
they  first  strike  a  plank.     Why  does  the  pitch  fall  when  the  plank  is 
pushed  further  against  the  saw  ? 

4.  Miners  entombed  by  a  fall  of  rock  or  by  an  explosion  have 
signaled  by  taps  on  a  pipe  or  by  pounding  on  the  rock.     How  does 
the  sound  reach  the  surface  ? 

5.  Two  Rookwood  vases  in  the  form  of  pitchers   with  slender 
necks  give  musical  sounds  when  one  blows  across  their  mouth.     Why 
does  the  larger  one  give  a  note  of  lower  pitch  than  the  smaller  ? 

6.  What  note  is  made  by  three  times  as  many  vibrations  as  c' 
(middle  C)  ? 

7.  If  cf  is  due  to  256  vibrations  per  second,  what  is  the  frequency 
of  g"  in  the  next  octave  ? 

8.  What  is  the  wave  length  of  g'  when  sound  travels  1130  feet 
per  second  ? 

9.  If  c'  has  264  vibrations  per  second,  how  many  has  a'? 

10.  When  sound  travels  1120  ft.  per  second,  the  wave  length  of  the 
note  given  by  a  fork  was  3.5  ft.  What  was  the  pitch  of  the  fork  ? 


VIBRATION   OF  STRINGS  167 


IX.   VIBRATION  OF  STRINGS 

207.  Manner  of  Vibration.  —  When  strings  are  used  to 
produce  sound,  they  are  fastened  at  their  ends,  stretched 
to  the  proper  tension,  and  are  made  to  vibrate  transversely 
by  drawing  a  bow  across  them,  striking  with  a  light  ham- 
mer as  in  the  piano,  or  plucking  with  the  fingers  as  in  the 
banjo,  guitar,  or  harp. 

208.  The  Sonometer.  —  The  sonometer  is  an  instrument 
for  the   study   of   the  laws  governing  the   vibration   of 
strings.     It  consists  of  a  thin  wooden  box,  across  which  is 
stretched  a  violin  string  or  a  thin  piano  wire  (Fig.  165). 


Fig.   165 

The  wires  pass  over  fixed  bridges,  A  and  1$,  near  the  ends, 
and  are  stretched  by  tension  balances  at  one  end.  They 
may  be  shortened  by  movable  bridges  <7,  sliding  along 
scales  under  the  wires. 

209.  Laws  of  Strings.  —  Stretch  two  similar  wires  on  the  so- 
nometer and  tune  to  unison  by  varying  the  tension.  Shorten  one  of 
them  by  moving  the  bridge  C  to  f,  f,  f ,  f ,  etc.  The  successive  inter- 
vals between  the  notes  given  by  the  two  wires  will  be  f ,  f ,  f ,  |,  etc. 
The  notes  given  by  the  wire  of  variable  length  are  those  of  the  major 
diatonic  scale.  Hence, 

The  frequency  of  vibration  for  a  given  tension  varies 
inversely  as  the  length. 

Starting  with  a  given  tension  and  the  strings  or  wires  in  unison, 
increase  the  stretching  force  on  one  of  them  four  times ;  it  will  now 


168  SOUND 

give  the  octave  of  the  other  with  twice  the  frequency.  Increase  the 
tension  nine  times;  it  will  give  the  octave  plus  the  fifth,  or  the 
twelfth,  above  the  other  with  three  times  the  frequency.  These  state- 
ments may  be  verified  by  dividing  the  comparison  wire  by  a  bridge 
into  halves  and  thirds,  so  as  to  put  it  in  unison  with  the  wire  of  vari- 
able tension.  Hence, 

When  the  length  is  constant,  the  frequency  varies  as 
the  square  root  of  the  tension. 

Stretch  equally  two  wires  differing  in  diameter  and  material,  that 
is,  in  mass  per  unit  length.  Bring  them  to  unison  with  the  movable 
bridge.  The  ratio  of  their  lengths  will  be  inversely  as  that  of  the 
square  roots  of  the  masses  per  unit  length.  Hence, 

The  length  and  tension  being  constant,  the  frequency 
varies  inversely  as  the  square  root  of  the  mass  per  unit 
length. 

210.  Applications.  —  In  the  piano,  violin,  harp,  and  other 
stringed  instruments,  the  pitch  of  each  string  is  determined 
partly  by  its  length,  partly  by  its  tension,  and  partly  by 
its  size  or  the  mass  of  fine  wire  wrapped  around  it.     The 
tuning  is  done  by  varying  the  tension. 

211.  Fundamental  Tone.  —  Fasten  one  end  of  a  silk  cord  about 
a  meter  long  to  one  prong  of  a  large  tuning  fork,  and  wrap  the  other 

end  around  a  wooden  pin 
inserted  in  an  upright  bar 
in  such  a  way  that  ten- 
sion can  be  applied  to  the 
cord  by  turning  the  pin. 
Set  the  fork  vibrating,  and 
adjust  the  tension  until  the 
cord  vibrates  as  a  whole 

(Fig.  166).      Arranged  in  this   way,   the  frequency   of   the  fork  is 

double  that  of  the  cord. 

The  experiment  shows  the  way  a  string  or  wire  vibrates 
when  giving  its  lowest  or  fundamental  tone.  A  body 


VIBRATION  OF  STRINGS 


169 


yields  its  fundamental  tone  when  vibrating  as  a  whole,  or 
in  the  smallest  number  of  segments  possible. 

212.  Nodes  and  Segments.  —  With  a  silk  cord  about  2  m.  long, 
and  mounted  as  in  the  last  experiment,  adjust  the  tension  until  the 
cord  vibrates  in  a  number  of 
parts,  giving  the  appearance 
of  a  succession  of  spindles 
of  equal  length  (Fig.  167). 
The  frequency  of  the  fork  is 
twice  that  of  each  spindle.  p.  ^ 

Stretch  a  wire  on  a  sonom- 
eter with  a  thin  slip  of  cork  strung  on  it.     Place  the  cork  at  one 
third,  one   fourth,   one   fifth,  or  one   sixth  part   of  the  wire  from 
one  end  ;  touch  it  lightly,  and  bow  the  shorter  portion  of  the  wire. 
The  wire  will    vibrate    in    equal   segments    (Fig.  168).      The  divi- 


Rg.  168 

sion  into  segments  may  be  made  more  conspicuous  by  placing  on  the 
wire,  before  bowing  it,  narrow  V-shaped  pieces  of  paper,  or  riders.  If, 
for  example,  the  cork  is  placed  at  one  fourth  the  length  of  the  wire, 
the  paper  riders  should  be  in  the  middle,  and  at  one  fourth  the  length 
from  the  other  end,  and  at  points  midway  between  these.  When  the 
wire  is  deftly  bowed,  the  riders  at  the  fourths  will  remain  seated,  and 
the  intermediate  ones  will  be  thrown  off.  The  latter  mark  points  of 
maximum,  and  the  former  those  of  minimum  vibration. 

The  ends  of   the  wire  and  the  intermediate  points  of 
least  motion  are  called  nodes;  the  vibrating  portions  be- 


170  SOUND 

tween  the  nodes  are  loops  or  segments;  and  the  middle 
points  of  the  loops  are  called  antinodes.  The  last  two  ex- 
periments illustrate  what  are  known  as  stationary  waves. 
They  result  from  the  interference  of  the  direct  system  of 
waves  and  those  reflected  from  the  fixed  end  of  the  wire. 
At  the  nodes  the  two  meet  in  opposite  phase  ;  at  the  anti- 
nodes  in  the  same  phase.  At  the  former  the  motion  is  re- 
duced to  a  minimum ;  at  the  latter  it  rises  to  a  maximum. 

213.  Overtones  in  Strings.  —  Stretch  two  similar  wires  on  the 
sonometer  and  tune  to  unison;  then  place  a  movable  bridge  at  the 
middle  of  one  of  them.  Set  the  longer  wire  in  vibration  by  plucking 
or  bowing  it  near  one  end.  The  tone  most  distinctly  heard  is  its 
fundamental.  Touch  it  lightly  at  its  middle  point;  instead  of  stop- 
ping the  sound,  a  tone  is  now 
heard  in  unison  with  that  given 


__^ ~"--^      P 

by  the  shorter  wire,  that  is,  an 
octave  higher  than  the   funda- 
mental and  caused  by  the  longer  wire  vibrating  in  halves  (Fig.  169). 
If  the  wire  be  again  plucked,  both  the  fundamental  and  the  octave 
may  be  heard  together. 

Touching  the  wire  one  third  from  the  end  brings  out  a  tone  in 
unison  with  that  given  by  the 
second  wire  reduced  to  one  third 
its  length  by  the  movable  bridge, 
that  is,  it  yields  a  tone  of  three 
times  the  frequency,  or  an  octave  and  a  fifth  higher  than  the  fundamen- 
tal. Figure  170  illustrates  the  manner  in  which  the  wire  is  vibrating. 

The  experiment  shows  that  a  wire  may  vibrate  not  only 
as  a  whole  but  at  the  same  time  in  parts,  yielding  a  com- 
plex note.  The  tones  produced  by  a  body  vibrating  in 
parts  are  called  overtones  or  partial  tones. 

214.  Harmonics.  —  If  the  frequency  of  vibration  of  the 
overtone  is  an  exact  multiple  of  the  fundamental,  it  is 
called  an  harmonic  partial  or  simply  an  harmonic.  In 


VIBRATION   OF  AIR  IN  PIPES 


171 


strings  the  overtones  are  usually  harmonics,  but  in  vibrat- 
ing plates  and  membranes  they  are  not. 

The  harmonics  are  named  first,  second,  third,  etc.,  in 
the  order  of  their  vibration  frequency.  The  frequency  of 
any  particular  harmonic  is  found  by  multiplying  that  of 
the  fundamental  by  a  number  one  greater  than  the  num- 
ber of  the  harmonic.  For  example,  the  frequency  of 
the  first  harmonic  of  c'  of  256  vibrations  per  second  is 
256  x  2  =  512 ;  that  of  the  second  is  256  x  3  =  768,  etc. 


X.     VIBRATION  OF  AIR  IN  PIPES 

215.   Air  as  a  Source  of  Sound.  —  In  the  use  of  the  res- 
onator we  saw  that  air  may  be  thrown  into  vibration  when 
it  is  confined  in  tubes  or  globes,  and  that  it  thus  becomes 
the  source  of    sound.     Such  a  body  of  air  may  be  set  /'  -' 
vibrating  in  two  ways :    by  a  vibrating  tongue  or  reed,       *J 


as  in  the  clarinet  (Fig.  171),  the  fish  horn,  etc.,  or  by  a 


' 


Fig.  171 


stream  of  air  striking  against  the  edge  of  an  opening 
in  the  tube,  as  in  the  whistle,  the  flute  (Fig.   172),  the 


V^  ,,,V,x    /=      „ 


Fig.  172 

I    organ  pipe,  etc.     In  several   pipe  or  wind  instruments  s>  > 

the  lips  of   the  player  act   as   reeds,  as   in    the   trumpet,  ^ 
|    trombone  (Fig.  173),  the  French  horn,  and  the  cornet. 


Fig.  173 


172 


SOUND 


Wind  instruments  may  be  classed  as  open  or  stopped 
pipes,  according  as  the  end  remote  from  the  mouthpiece 
is  open  or  closed. 

216.    Fundamental  of  a  Closed  Pipe. — Let  the   tall   jar   of 
Fig.  174  be  slowly  filled  with  water  until  it  responds  strongly  to  a 
c'  fork,  for  example.     The  length  of  the  column 
of  air  will  be  about  13  in.  or  one  fourth  of  the 
wave  length  of  the  note. 

When  the  prong  at  a  moves  to  Z>,  it  makes  half 
a  vibration,  and  generates  half  a  sound  wave.  It 
sends  a  condensed  pulse  down  the  tube  AB,  and 
this  pulse  is  reflected  from  the  water  at  the 
bottom.  Now,  if  AB  is  one  fourth  a  wave  length, 
the  distance  down  and  back  is  one  half  a  wave 
length,  and  the  pulse  will  return  to  A  at  the  in- 
stant when  the  prong  begins  to  move  from  b  back 
to  a,  and  to  send  a  rarefaction  down  AB.  This 
in  turn  will  run  down  the  tube  and  back,  as  the 
prong  completes  its  vibration;  the  co-vibration 


Fig.  174 


is  then  repeated  indefinitely,  the  tube  responds  to  the  fork,  and  its 
length  is  one  quarter  of  the  wave  length.     Hence, 

The  fundamental  of  a  closed  pipe  is  a  note  whose  wave 
length  is  four  tim,es  the 
length  of  the  pipe. 

217.   Laws  for  Columns 

of  Air.  —  Set  vertically  in 
a  wooden  base  eight  glass 
tubes  each  about  25  cm.  long 
and  2  cm.  in  diameter  (Fig. 
175).  Pour  in  them  melted 
paraffin  to  close  the  bottom. 
A  musical  note  may  be  pro- 
duced by  blowing  a  stream 
of  air  across  the  top  of  each 
tube.  From  the  confused  flutter  made  by  the  air  striking  the  edge 
of  the  tube,  the  column  of  air  selects  for  reinforcement  the  frequency 


Fig.  175 


VIBRATION   OF  AIR  IN  PIPES 


173 


corresponding  to  its  own  rate.  Hence  the  pitch  may  be  varied  by  pour- 
ing in  water.  Adjust  all  the  tubes  with  water  until  they  give  the  eight 
notes  of  the  major  diatonic  scale.  The  measured  lengths  of  the  col- 
umns of  air  will  be  found  to  be  nearly  as  1,  f,  f ,  £,  f,  f,  T83,  |.  The  notes 
emitted  have  the  frequencies  1,  f,  |,  f,  |,  |,  Y,  2  (§  204).  Hence, 

The  frequency  of  a  vibrating  column  of  air  is  inversely 
as  its  length. 

This  is  the  principle  employed  in  playing  the  trombone. 

Blow  gently  across  the  end  of  an  open  tube  30  cm.  long  and  about 
2  cm.  in  diameter  and  note  the  pitch.     Take  another  tube  of  the 
same  diameter  and  15  cm.  long ;  stop  one  end  by  pressing 
it  against  the  palm  of  the  hand,  and  sound  it  by  blowing 
across  the  open  end.    The  pitch  of  the  closed  pipe  will  be 
the  same  as  that  of  the  open  one.     The  experiment  may 
be  varied  by  comparing  the  notes  obtained  by  the  shorter 
pipe  when  open  and  when  closed  at  one  end  ;  the  former 
will  be  an  octave  higher  than  the  latter.     Hence, 

For  the  same  frequency,  the  open  pipe  is  twice 
the  length  of  the  stopped  one. 

The  length  -of  the  open  pipe  is,  therefore,  half 
the  wave  length  of  the  fundamental  note  in  air. 

218.  State  of  the  Air  in  a  Sounding  Pipe.  —  Em- 
ploying an  open  organ  pipe,  preferably  with  one  glass 
side  (Fig.  176),  lower  into  it  a  miniature  tambourine 
about  3  cm.  in  diameter  and  covered  with  fine  sand, 
while  the  pipe  is  sounding  its  fundamental  note.  The 
sand  will  be  agitated  most  at  the  ends  of  the  pipe  and 
very  little  at  the  middle.  There  is,  therefore,  a  node  at 
the  middle  of  an  open  pipe.  A  node  is  a  place  of  least 
motion  and  greatest  change  of  density;  an  antinode  is  a 
place  of  greatest  motion  and  least  change  of  density. 
The  closed  end  of  a  pipe  is  necessarily  a  node,  and  the  Fig.  176 
Open  end  an  antinode.  Hence, 

In  an  open  pipe,  for  the  fundamental  tone,  there  is 
a  node  at  the  middle  and  an  antinode  at  each  end;  in 


174 


SOUND 


the  stopped  pipe,  there  is  a  node  at  the  closed  end  and 
an  antinode  at  the  other  end. 

219.  Overtones  in  Pipes.  — Blow  across  the  open  end  of  a  glass 
tube  about  75  cm.  long  and  2  cm.  in  diameter.  A  variety  of  tones  of 
higher  pitch  than  the  fundamental  may  be  obtained  by  varying  the 
force  of  the  stream  of  air.  ' 

These  tones  of  higher  pitch  than  the  fundamental  are 
overtones  ;  they  are  caused  by  the  column  of  air  vibrating 
in  parts  or  segments  with  intervening  nodes. 

Open  pipes  give  the  complete  series  of  overtones,  with  fre- 
quencies 2,  3,  4,  &•>  etc.  times  that  of  the  fundamental. 

In  stopped  pipes  only  those  overtones  are  possible  whose 
frequencies  are  3,  J,  7,  etc.  times  that  of  the  fundamental. 
Briefly,  the  reason  is  that  with  a  node  at  one  end  and  an 
antinode  at  the  other,  the  column  of  air  can  divide  into 
an  odd  number  of  equal  half  segments  only. 

It  follows  that  the  notes  given  by  open  pipes  differ  in 
quality  from  those  of  closed  pipes. 


XL     GRAPHIC  AND  OPTICAL  METHODS 

220.  Kecord  of  Vibrations.  —  Graphic  methods  of  study- 
ing sound  are  of  service 
in  determining  the  fre- 
quency of  vibration. 
Figure  177  shows  a 
practical  device  for 
this  purpose.  A  sheet 

'KULW^S  of    PaPer    is    wrapped 

s==^^5r     \  '"-'1§P^a  around  a  metal  cylin- 

der, and  is  then  smoked 
with    lampblack.       A 
Fig   177  large  fork  is  securely 


GEAPHIC  AND   OPTICAL  METHODS 


175 


mounted,  so  that  a  light  style  attached  to  one  prong 
touches  the  paper  lightly.  The  cylinder  is  mounted  on  an 
axis,  one  end  of  which  has  a  screw  thread  cut  in  it,  so 
that  when  the  cylinder  turns  it  also  moves  in  the  direction 
of  its  axis.  The  beats  of  a  seconds  pendulum  may  be 
marked  on  the  paper  by  means  of  electric  sparks  between 
the  style  and  the  cylinder.  The  number  of  waves  between 
successive  marks  made  by  the  spark  is  equal  to  the  fre- 
quency of  the  fork. 

221.  Manometric  Flames.  —  A  square  box  with  mirror 
faces  is  mounted  so  as  to  turn  around  a  vertical  axis 
(Fig.  178).  In  front 
of  the  revolving  mir- 
rors is  supported  a 
short  cylinder,  which 
is  divided  into  two 
shallow  chambers  by 
a  partition  of  gold- 
beater's skin  or  thin 
rubber.  Illuminat- 
ing gas  is  admitted 
to  the  compartment 
on  the  right  through 
the  tube  with  a  stop- 
cock, and  burns  at 
the  small  gas  jet  on 

Ri  v    \  7R 

the  little  tube  run- 
ning into  this  same  compartment.     The  speaking  tube  is 
connected  to  the  compartment  on  the  other  side  of  the 
flexible  partition. 

When   the   mirrors   are   turned   by  twirling  with  the 
thumb  and  finger  the  milled  head  at  the  top,  the  image 


176 


SOUND 


Fig.  179 


of  the  gas  jet  is  drawn  out  into  a  smooth  band  of  light. 
Any  pure'  tone  at  the  mouthpiece  produces  alternate  com- 
pressions and  rarefactions  in  both  chambers  separated  by 

the  membrane,   and  these 

aid  and  retard  the  flow  of 
gas  to  the  burner.  The 
flame  changes  shape  and 
flickers,  but  its  vibrations 
are  too  rapid  to  be  seen 
directly.  But  if  it  is  ex- 
amined by 
reflection 
from  t  he 
rotating 
mirrors,  its 
image  is  a 
serrated  band  (Fig.  179). 

Koenig  fitted  three  of  these  little  cap- 
sules with  jets  to  the  side  of  an  open 
organ  pipe  (Fig.  180),  the  membrane  on 
the  inner  side  of  the  gas  chamber  form- 
ing part  of  the  wall  of  the  pipe.  When 
the  pipe  is  blown  so  as  to  sound  its  funda- 
mental tone,  the  middle  point  is  a  node 
with  the  greatest  variations  of  pressure 
in  the  pipe,  and  the  flame  at  that  point  is 
more  violently  agitated  than  at  the  other 
two,  giving  in  the  mirrors  the  top  band  of 
Fig.  179.  By  increasing  the  air  blast,  the 
fundamental  is  made  to  give  way  to  the 
first  overtone;  the  two  outside  jets  then 
vibrate  most  strongly,  and  give  the  second 
band  in  the  figure,  with  twice  as  many  Fig- 


QUESTIONS  AND  PROBLEMS  177 

tongues  of  flame  as  in  the  image  for  the  fundamental. 
The  third  band  may  be  obtained  by  adjusting  the  air 
pressure  so  that  both  the  fundamental  and  the  first  over- 
tone are  produced  at  the  same  time.  This  same  figure 
may  be  obtained  by  singing  into  the  mouthpiece  or  funnel 
of  Fig.  178  the  vowel  sound  o  on  the  note  -B,  showing 
that  this  vowel  sound  is  composed  of  a  fundamental  and 
its  octave. 

222.   Kundt's  Dust  Tube.  —  The  division  of   a  resonant 
pipe  into  segments  may  be  beautifully  shown  by  means 


Fig.  181 

of  a  glass  tube  about  2  cm.  in  diameter  and  40  cm.  long. 
One  end  is  closed,  and  a  common  whistle  is  attached  to 
the  other  (Fig.  181).  Within  the  tube  is  evenly  sifted  a 
little  dry  cork  dust,  or  amorphous  silica.  When  the  whis- 
tle is  blown,  the  powder  is  caught  up  by  the  moving  air 
at  the  antinodes,  and  settles  down  in  small  circles  at  the 
nodes.  At  the  same  time,  between  the  nodes  it  is  divided 
into  thin,  airy  segments,  with  vertical  divisions,  the  agita- 
tion being  sufficient  to  support  the  dust  in  opposition  to 
gravity.  The  subdivision  changes  when  the  blast  of  air 
is  increased  to  give  overtones. 

Questions  and  Problems 

1.  Name  three  ways  in  which  musical  sounds  may  differ. 

2.  Pianos  are  made  so  that  the  hammers  strike  the  wires  near  one 
end  and  not  in  the  middle.     Why? 

3.  Why  does  the  pitch  of  the  sound  made  by  pouring  water  into 
a  tall  cylindrical  jar  rise  as  the  jar  fills? 


178  SOUND 

4.  What  effect  does  a  rise  of  temperature  have  on  the  pitch  of  a 
given  organ  pipe  ?     Explain. 

5.  If  the  pipes  of  an  organ  are  correctly  tuned  at  a  temperature 
of  40°  F.,  will  they  still  be  in  tune  at  90°  F.  ?    Explain. 

6.  The  tones  of  three  bells  form  a  major  triad.     One  of  them 
gives  a  note  a  of  220  vibrations  per  second,  and  its  pitch  is  between 
those  of  the  other  two.     What  are  the  frequencies  of  the  three  bells, 
and  what  is  the  note  given  by  the  highest  ? 

7.  How  much  must  the  tension  of  a  violin  string  be  increased  to 
raise  its  pitch  a  fifth  (§  203)  ? 

8.  If  the  E  string  of  a  violin  is  40  cm.  long,  how  long  must  a 
similar  one  be  to  give  G  ? 

9.  The  vibration  frequency  of  two  similar  wires  100  cm.  long  is 
297.     How  many  beats  per  second  will  be  given  by  the  two  wires 
when  one  of  them  is  shortened  one  centimeter? 

10.  Two  c'  forks  gave  5  beats  per  second  when  one  of  them  was 
weighted  with  bits   of   sealing   wax.      Find  the   frequency  of  the 
weighted  fork. 

11.  What  will  be  the  length  of  a  stopped  organ  pipe  to  give  c1  of 
256  vibrations  per  second  when  the  temperature  of  the  air  is  20°  C  ? 

12.  Calculate  the  length  of  an  open  organ  pipe  whose  fundamental 
tone  is  one  of  32  vibrations  per  second,  and  the  temperature  of  the  air 
is  20°  C. 

13.  An  open  organ  pipe  sounds  c'  (256)  ;  what  notes  are  its  two 
lowest  overtones  ? 

14.  What  is  the  frequency  of  an  8-foot  stopped  pipe  when  the 
velocity  of  sound  is  1120  ft.  per  second? 

15.  Two  open  organ  pipes  2  ft.  in  length  are  blown  with  air  at  a 
temperature  of  15°  and  20°  C.,  respectively.     How  many  beats  do  they 
give  per  second  ? 

16.'  When  the  temperature  of  the  air  is  such  that  the  velocity  of 
sound  is  1105  ft.  per  second,  what  will  be  the  frequency  of  the  funda- 
mental note  produced  by  blowing  across  one  end  of  a  tube  12.75  in. 
long,  the  other  end  being  closed  ?  What  wil)  be  the  frequency  of  its 
first  overtone  ? 


i 


CHAPTER   VIII 

LIGHT 
I.    NATURE  AND  TRANSMISSION  OF  LIGHT 

223.  The  Ether.  —  Exhaust  the  air  as  far  as  possible  from  a  glass 
bell  jar.     Place  a  candle  on  the  far  side  of  the  jar;  it  will  be  seen  as 
clearly  before  the  air  has  been  let  into  the  bell  jar  as  after. 

It  is  obvious  that  the  medium  conveying  light  is  not 
the  air  and  it  must  be  something  that  exists  even  in  a 
vacuum.  Physicists  have  agreed  to  call  this  medium 
the  ether.  It  exists  everywhere,  even  penetrating  between 
the  molecules  of  ordinary  matter.  Little  is  known  about 
its  nature  and  the  exact  way  in  which  light  travels  through 
it,  but  it  is  generally  agreed  that  light  is  a  wave  motion  in 
the  ether  and  that  the  vibrations  are  not  longitudinal  as  in 
sound  waves,  but  transverse  (§  174). 

The  theory  that  light  is  a  wave  motion  in  the  ether 
was  proposed  by  Huyghens,  a  Dutch  physicist,  in  1678; 
Fresnel,  a  French  physicist,  showed  that  the  disturbance 
must  be  transverse ;  and  Maxwell  modified  the  theory  to 
the  effect  that  these  disturbances  are  probably  not  trans- 
verse physical  movements  of  the  ether,  but  transverse 
alterations  in  its  electrical  and  magnetic  conditions. 

224.  Transparent  and  Opaque  Bodies. — When  light  falls 
on  a  body,  a  part  of  it  is  reflected,  a  part  passes  through 
or  is  transmitted,  and  the  rest  is  absorbed.      A  body  is 
transparent  when  it  allows  light  to  pass  through  it  with 
so   little   loss   that   objects   can    be   easily   distinguished 
through  it,  as  glass,  air,  pure  water.      Translucent  bodies 

179 


180 


LIGHT 


transmit  light,  but  so  imperfectly  that  objects  cannot  be 
seen  distinctly  through  them,  as  horn,  oiled  paper,  very 
thin  sheets  of  metal  or  wood.  Other  bodies,  such  as 
blocks  of  wood  or  iron,  transmit  no  light,  and  these  are 
opaque.  No  sharp  line  of  separation  between  these  classes 
can  be  drawn ;  the  classification  is  one  of  degree.  Water 
when  deep  enough  cuts  off  all  light ;  the  bottom  of  the 
deep  ocean  is  dark.  Stars  which  are  invisible  at  the  foot 
of  a  mountain  are  often  visible  at  the  top. 

225.    Speed  of  Light.  — Previous  to  the  year  1676  it  was 
believed   that   light   traveled   infinitely  fast,  because   no 

one  had  found  a 
way  to  measure 
so  great  a  veloc- 
ity. But  in  that 
year  Roemer,  a 
young  Danish  as- 
tronomer, made 
the  very  impor- 
tant discovery 
that  light  travels 
with  finite  speed. 
Roemer  was  en- 
gaged at  the 
Paris  Observatory  in  observing  the  eclipses  of  the  inner 
satellite  or  moon  of  the  planet  Jupiter.  At  each  revolu- 
tion of  the  moon,  M  (Fig.  182)  in  its  orbit  round  the 
planet  J",  it  passes  into  the  shadow  of  the  planet  and 
becomes  invisible  from  the  earth  at  E,  or  is  eclipsed.  By 
comparing  his  observations  with  much  earlier  recorded 
ones,  Roemer  found  that  the  mean  interval  of  time  be- 
tween two  successive  eclipses  was  42.5  hours.  From  this 


Fig.  182 


James  Clerk-Maxwell  (1831-1879)  was  a  remarkable  physi- 
cist and  mathematician.  He  was  born  in  Edinburgh  and  studied 
in  the  University  of  that  city.  Later  he  attended  the  University 
of  Cambridge,  graduating  from  there  in  1854.  In  1856  he  be- 
came professor  of  natural  philosophy  at  Marischal  College,  Aber- 
deen, arid  in  1860  professor  of  physics  and  astronomy  at  King's 
College,  London.  In  1871  he  was  appointed  professor  of  experi- 
mental physics  in  Cambridge.  His  contributions  to  the  kinetic 
theory  of  gases,  the  theory  of  heat,  dynamics,  and  the  mathemati- 
cal theory  of  electricity  and  magnetism  are  imperishable  monu- 
ments to  his  great  genius  and  wonderful  insight  into  the  mysteries 
of  nature. 


NATURE  AND   TRANSMISSION   OF  LIGHT          181 

it  was  easy  to  calculate  in  advance  the  time  at  which  suc- 
ceeding eclipses  would  occur.  But  when  the  earth  was 
going  directly  away  from  Jupiter,  as  at  JEV  the  eclipse 
interval  was  found  to  be  longer  than  anywhere  else ;  and 
at  Ey  across  the  earth's  orbit  from  Jupiter,  each  eclipse 
occurred  about  1000  sec.  later  than  the  predicted  time. 
To  account  for  this  difference  Roemer  advanced  the 
theory  that  this  interval  of  1000  sec.  is  the  time  taken 
by  light  to  pass  across  the  diameter  of  the  earth's  orbit. 
This  gave  for  the  speed  of  light  309  million  meters,  or 
192,000  mi.  per  second. 

Later  determinations  in  our  own  country  by  Michelson 
and  Newcomb  show  that  the  speed  of  light  is  299,877  km., 
or  186,337  mi.  per  second. 

226.  Direction  of  Propagation.  —  Place  a  sheet-iron  cylinder  over 
a  strong  light,  such  as  a  Welsbach  gas  lamp,  in  a  darkened  room. 
The  cylinder  should  have  a  small  hole  opposite  the  light.     Stretch  a 
heavy  white   thread  in  the  light  streaming  through  the   aperture. 
When  the  thread  is  taut  it  is  visible  throughout  its  entire  length,  but 
if  permitted  to  sag  it  becomes  invisible. 

The  experiment  shows  that  light  travels  in  straight  lines. 
It  will  appear  later  that  this  is  true  only  when  the  medium 
through  which  light  passes  has  the  same  physical  proper- 
ties in  all  directions. 

227.  Ray,  Beam,  Pencil.  — Light  is  propagated  outward 
from  the  luminous  source  in  concentric  spherical  waves, 
as  sound  waves  in  air  are  from  a  sonorous  body.    Rays  are 
the  radii  of  these  spherical  waves,  and  they  are,  therefore, 
normal  (perpendicular)  to  them.     They  mark  the  direc- 
tion of  propagation. 

When  the  source  of  light  is  at  a  great  distance,  the  rays 
incident  on  any  surface  are  sensibly  parallel.  A  number 
of  parallel  rays  form  a  beam  of  light.  For  example,  in  the 


182 


LIGHT 


case  of  light  from  the  sun  or  stars,  the  distance  is  so  great 
that  the  rays  are  sensibly  parallel.  Rays  of  light  pro- 
ceeding outward  from  a  point  form  a  diverging  pencil; 
rays  proceeding  toward  a  point,  a  converging  pencil. 


Fig.  183 

228.  Shadows.  —  Place  a  ball  between  a  lighted  lamp  and  a  white 
screen.  From  a  part  of  this  screen  the  light  will  be  wholly  cut  off, 
and  surrounding  this  area  is  one  from  which  the  light  is  excluded  in 
part.  If  three  small  holes  be  made  in  the  screen,  one  where  it  is 
darkest,  one  in  the  part  where  it  is  less  dark,  and  one  in  the  lightest 
part,  it  will  be  found  when  one  looks  through  them  that  the  flame  of 
the  lamp  is  wholly  invisible  through  the  first,  a  part  of  it  is  visible 
through  the  second,  and  the  whole  flame  through  the  third. 


Fig.  184 


The  space  behind  the  opaque  object  from  which  the 
light  is  excluded  is  called  the  shadow.  The  figure  on 
the  screen  is  a  section  of  the  shadow.  The  darkest  part 
of  the  shadow,  called  the  umbra,  is  caused  by  the  total 
exclusion  of  the  light  by  the  opaque  object ;  the  lighter 
part,  caused  by  its  partial  exclusion,  is  called  the  penumbra. 

When  the  source  of  light  is  a  point  L  (Fig.  188),  the 
shadow  will  be  bounded  by  a  cone  of  rays,  ALB,  tangent 
to  the  object,  and  will  have  only  one  part,  the  umbra. 
When  the  source  of  light  is  an  area,  such  as  LL  (Fig.  184), 


NATURE  AND   TRANSMISSION  OF  LIGHT 


183 


the  space  ABDO  behind  the  opaque  body  receives  no  light, 
and  the  parts  between  AC  and  AC9,  and  between  BD  and 
BD',  receives  some  light,  the  amount  increasing  as  AC9 
and  BD9  are  approached.  From  these  figures  the  cases 
when  the  luminous  body  is  larger  than  the  opaque  body, 
and  when  it  is  of  the  same  size,  may  be  understood  and 
illustrated  by  the  student. 

229.   Images  by  Small  Openings.  —  Support  two  sheets  of  card- 
board (Fig.  185),  in  vertical  planes.     In  the  center  of  one  cut  a  hole 
2  mm.  square,  and 
place  in  front  of  it 
a  lighted  candle  at 
a   distance   of    20 
or  25  cm.     An  in- 
verted image  of  the 
flame  of  the  candle 
will  appear  on  the 
other  sheet.     If  a 
second  small  open- 
ing be  made  near 
the  first,  a  second 
image  will  be  ob- 
tained not  coincid- 
ing   exactly   with 
the  first  one.     The  shape  of  the  opening,  so  long  as  it  is  small,  has  no 
effect  on  the  image.    With  a  larger  opening  the  image  gains  in  bright- 
ness but  loses  in  distinctness. 

Every  point  of  the  candle  flame  is  the  vertex  of  a  cone 
of  rays,  or  a  diverging  pencil,  passing  through  the  opening 
and  forming  an  image  of  it  on  the  screen.  These  numer- 
ous pictures  of  the  opening  overlap  and  form  a  picture  of 
the  flame,  and  the  number  at  any  one  place  determines 
the  brightness.  The  edge  of  the  image  will  therefore  be 
less  bright  than  other  portions.  In  the  case  of  a  large 
opening,  the  overlapping  of  the  images  of  the  aperture 


Fig.  185 


184 


LIGHT 


destroys  all  resemblance  between  the  image  and  the  ob- 
ject, the  resulting  image  having  the  shape  of  the  aperture. 

230.  Illustrations.  —  The  pinhole  camera  is  an  applica- 
tion of  the  foregoing  principle.     It  consists  of  a  small 
box,  blackened  within,  and  provided  with  a  small  opening 
in  one  face ;  the  light  passes  through  this  and  forms  an 
image  on  the  sensitized  plate  placed  on  the  opposite  side. 
When  the  sun  shines  through  the  small  chinks  in   the 
foliage  of  a  tree,  a  number  of  round  or  oval  spots  of  light 
may  be  Seen  on  the  ground.     These  are  images  of  the  sun. 
During  a  partial  solar  eclipse  such  figures  assume  a  cres- 
cent shape. 

II.     PHOTOMETRY 

231.  Law  of  Intensity.  — The  intensity  of  illumination  is 
the  quantity  of  light  received  on  a  unit  of  surface.     Every- 
day experience  shows  that  it  varies,  not  only  with  the 
source  of  the  light,  but  also  with  the  distance  at  which 
the  source  is  placed. 

C 


Fig.  186 

Cut  three  cardboard  squares,  4,  8,  and  12  cm.  on  a  side  respectively, 
and  mount  them  on  supports  (Fig.  186).  The  centers  of  these  screens 
should  be  at  the  same  distance  above  the  table  as  the  source  of  light. 
Use  a  Welsbach  gas  lamp  with  an  opaque  chimney  having  a  small 
opening  opposite  the  center  of  the  light,  and  set  it  99  cm.  from  the 


PHOTOMETRY 


185 


largest  screen.  Place  the  medium- sized  screen  so  that  it  exactly  cuts 
off  the  light  from  the  edges  of  the  largest.  In  like  manner  place  the 
smallest  screen  with  respect  to  the  intermediate  one.  If  these  screens 
are  placed  with  care,  it  will  be  found  that  their  distances  from  the 
light  are  33,  66,  and  99  cm.  respectively,  or  as  1  :  2  :  3.  Now  as  each 
screen  exactly  cuts  off  the  light  from  the  one  next  farther  away,  it 
follows  that  each  receives  the  same  amount  of  light  from  the  source 
when  the  light  is  not  intercepted.  The  surfaces  of  the  screens  are  as 
1:4:9,  and  hence  the  quantity  of  light  per  unit  of  surface  must  be 
inversely  as  1  :  4  :  9,  the  squares  of  1,  2,  and  3  respectively. 

This  experiment  shows  that  the  intensity  of  illumination 
varies  inversely  as  the  square  of  the  distance  from  the  source 
of  light.  If  the  medium  is  such  as  to  absorb  some  of  the 
light,  the  decrease  in  intensity  is  greater  than  that  ex- 
pressed by  the  law  of  inverse  squares. 

232.  The  Bunsen  Photometer. — A  photometer  is  an  in- 
strument  for  comparing  the  intensity  of  one  light  with  that 


Fig.  187 

of  another.  The  principle  applied  is  a  consequence  of  the 
.law  of  the  intensity  of  illumination;  it  is  that  the  ratio  of 
the  intensities  of  two  lights  is  equal  to  the  square  of  the 
ratio  of  the  distances  at  which  they  give  equal  illumination. 
In  the  Bunsen  photometer  a  screen  of  paper  A  (Fig.  187), 


186  LIGHT 

having  a  translucent  spot  made  by  applying  a  little  hot 
paraffin,  is  supported  on  a  graduated  bar  between  a 
standard  candle  B  and  the  light  O  to  be  compared  with  it. 
An  old  but  imperfect  standard  candle  is  the  light  emitted 
by  the  sperm  candle  of  the  size  known  as  "  sixes,"  when 
burning  120  grains  per  hour.  The  photometer  screen  is 
usually  enclosed  in  a  box  open  toward  the  two  lights,  and 
back  of  it  are  two  mirrors  placed  with  their  reflecting 
sides  toward  each  other  in  the  form  of  a  V,  so  that  the 
observer  standing  by  the  side  of  A  can  see  both  sides  of 
the  screen  by  reflection  in  the  mirrors.  The  position  of 
A  or  of  B  may  then  be  adjusted  until  both  sides  of  the 
screen  look  alike.  Then  the  intensity  of  Q  is  to  the  in- 
tensity of  B  as  AC2  is  to  Alf. 

Questions  and  Problems 

1.  What  is  the  cause  of  an  eclipse  of  the  moon? 

2.  What  is  the  cause  of  an  eclipse  of  the  sun  ? 

3.  Why  is  the  duration  of  a  lunar  eclipse  greater  than  that  of  a 
solar  eclipse  ? 

4.  Atropine  placed  in  the  eye  enlarges  the  pupil.     Why  is  vision 
then  less  distinct  ? 

5.  An  illuminated  vertical  object  4  feet  long  is  at  a  distance  of 
12  feet  from  a  shutter  in  which  there  is  a  minute  hole.     Inside  is  a 
vertical  screen  4  feet  from  the  small  aperture.     How  long  is  the  image 
of  the  illuminated  object  given  by  the  small  aperture? 

6.  The  pupil  of  the  eye  of  a  cat  is  elliptical  rather  than  round. 
Is  the  form  of  the  image  on  the  retina  of  the  eye  affected  by  the  shape 
of  the  pupil? 

7.  If  a  yardstick  standing  vertically  casts  a  shadow  2  ft.  long, 
how  high  is  a  flagpole  that  at  the  same  time  casts  a  shadow  100  ft.  long? 

8.  The  following  data  were   obtained  in  measuring  the  candle 
power  of  a  Welsbach  gas  flame :     Distance  of  standard  candle  from 
photometer  disk,  15  cm.;   distance  of  lamp,  100  cm.     What  is  the 
candle  power  of  the  gas  flame  ? 


REFLECTION   OF  LIGHT 


187 


9.  Two  lamps  of  equal  candle  power  are  placed  100  m.  apart. 
Where  must  a  screen  be  placed  between  them  so  that  one  side  re- 
ceives four  times  as  much  light  as  the  other  ? 

10.  Two  electric  lights  of  16  and  64  candle  power,  respectively, 
are  placed  in  front  of  a  picture  so  as  to  illuminate  it  equally.  The 
16  candle  power  lamp  is  10  ft.  from  the  picture ;  at  what  distance  is 
the  other  one  ? 

III.    REFLECTION  OF  LIGHT 

233.  Regular  Reflection.  —  When  a  beam  of  light  falls  on 
a  polished  plane  surface,  the  greater  part  of  it  is  reflected  e 
in  a  definite  direction.  This 
reflection  is  known  as  regu- 
lar reflection.  In  Fig.  188  a 
beam  of  light  IB  is  incident 
on  the  plane  mirror  B  and  is 
reflected  as  BE.  IB  is  the 
incident  beam,  BR  is  the  re- 
flected beam,  the  angle  IBP 
between  the  incident  beam 
and  the  normal  (perpendicu- 
lar) to  the  reflecting  surface 
is  the  angle  of  incidence,  and 
the  angle  PBR  between  the 


.V 


Fig.  188 
reflected  beam  and  the  normal  is  the  angle  of  reflection. 


234.    Law  of  Reflection.  —  On  a  semicircular  board  are  mounted 
two  arms,  pivoted  at  the  center  of  the  arc  (Fig.  189).     One  arm 

.  carries  a  vertical 
rod  P,  and  the 
other  a  paper  tube 
T  with  parallel 
threads  stretched 
across  a  diameter 
at  each  end.  A 
Fig.  189  plane  mirror  M 


188 


LIGHT 


is  mounted  at  the  center  of  the  semicircle,  with  its  reflecting  surface 
parallel  to  the  diameter  at  the  ends  of  the  arc.  On  the  edge  of  the 
semicircle  is  a  scale  of  equal  parts  with  the  zero  on  the  normal  to  the 
mirror.  Place  the  arm  P  in  any  desired  position  and  move  the  arm 
T  until  the  image  of  the  rod  in  the  mirror  is  exactly  in  line  with  the 
two  thr-eads.  The  scale  readings  will  show  that  the  two  arms  make 
equal  angles  with  the  normal  to  the  mirror.  Hence, 

The  angle  of  reflection  is  equal  to  the  angle  of  inci- 
dence, and  the  two  angles  lie  in  the  same  plane. 

235.  Diffused  Reflection.  —  Cover  a  large  glass  jar  with  a  piece 
of  cardboard,  in  which  is  a  hole  about  1  cm.  in  diameter.  Fill  the  jar 
with  smoke,  and  reflect  into  it  through  the  hole  in  the  cover  a  beam 
of  sunlight.  The  whole  of  the  interior  of  the  jar  will  be  illuminated. 

The  small  particles  of  smoke  floating  in  the  jar  furnish 
a  great  many  reflecting  surfaces  ;  the  light  falling  on 
them  is  reflected  in  as  many  directions.  The  scattering 
of  light  by  uneven  or  irregular  surfaces  is  diffused  reflec- 
tion. 

To  a  greater  or  less  extent  all  reflecting  surfaces  scatter 
light  in  the  same  way  as  the  smoke  particles.  Figure  190 

illustrates  in  an  ex- 
aggerated way  the 
difference  between 
a  perfectly  smooth 
surface  and  one 
somewhat  uneven. 
It  is  by  diffused 
reflection  that  objects  become  visible  to  us.  Perfect  re- 
flectors would  be  invisible  ;  it  is  almost  impossible  to  see 
the  glass  of  a  very  perfectly  polished  mirror.  The  trees, 
the  ground,  the  grass,  and  particles  floating  in  the  air  re- 
flect the  light  from  the  sun  in  every  direction,  and  thus 
fill  the  space  about  us  with  light.  If  the  air  were  free 


Fig.  190 


REFLECTION  OF  LIGHT 


189 


from  all  floating  particles  and  gases,  the  sky  would  be 
dark  in  all  directions,  except  in  the  direction  of  the  sun 
and  the  stars.  This  conclusion  is  confirmed  by  aeronauts 
who  have  reached  very  high  altitudes,  where  there  was 
almost  a  complete  absence  of  floating  particles. 

236.  Image  of  an  Object  in  a  Plane  Mirror.  —  Any  smooth 
reflecting  surface  is  called  a  mirror.      A  plane  mirror  is 
one  whose  reflecting  surface  is  a  plane.     A  spherical  mir- 
ror is   one   whose   reflecting   surface   is   a   portion  of   a 
sphere. 

Support  a  pane  of  clear  window  glass  in  a  vertical  position,  and 
place  a  red-colored  lighted  candle  back  of  it.  Place  a  white  candle  in 
front  but  not  lighted.  Move  the  unlighted  candle  until  its  image  in 
the  glass  as  a  mirror  coincides  exactly  with  the  lighted  candle  seen 
through  the  glass.  The  distance  of  the  two  candles  from  the  mirror 
will  be  the  same. 

The  image  of  an  object  in  a  plane  mirror  is  a  virtual 
image  because  the  light  only  apparently  comes  from  it. 
This  image  is  of  the  same  size  as  the  object  and  is  as  far 
back  of  the  mirror  as  the  object  is  in  front. 

237.  Geometrical  Position  of  the  Image  of  a  Point.  —  Let 
A  (Fig.   191)  be  a  luminous  point  in  front  of  a  plane 
mirror  MN.     Any  ray  AB  in- 
cident   on   the   mirror   is   re- 
flected  in  the  direction  BD, 

making  the  angle  of  reflection 
FBD  equal  to  the  angle  of 
incidence  FBA.  In  like  man- 
ner, a  second  ray  AC  is  re- 
flected along  CE.  From  A 
drop  a  perpendicular  AK  to 
the  reflecting  surface  and  pro-  Fig.  191 


190 


LIGHT 


duce  it  behind  the  mirror.  Since  the  incident  and  re- 
flected rays  AB  and  BD  are  in  the  same  plane  with 
the  normal  AK,  it  follows  that  BD  produced  meets  the 
perpendicular  in  some  point  A'.  The  point  A'  is  as 
far  behind  the  mirror  as  A  is  in  front.1  The  ray  CE 
produced  backwards  also  meets  the  perpendicular  at  A'. 
But  AB  and  AC  are  any  two  rays  incident  on  the  mirror. 
It  follows  that  all  rays  from  A,  incident  on  MN^  are 
reflected  from  MN  as  if  they  came  from  a  point  as  far 
back  of  the  mirror  as  A  is  in  front.  Hence  the  eye  placed 
in  the  region  of  D  or  E  will  receive  the  reflected  rays  as 
if  they  came  from  A1.  The  point  A  is  the  image  of  A  in 
the  mirror  MN,  and  it  is  a  virtual  image  because  the  light 
only  apparently  or  virtually  comes  from  it.  Therefore, 
the  image  of  a  point  in  a  plane  mirror  is  virtual  and  is 
as  far  back  of  the  mirror  as  the  point  is  in  front  of  it. 

238.   Construction  for  an  Image  in  a  Plane  Mirror.  —  As 

the  image  of  an  object  is  composed  of  the  images  of  its 

points,  the  image  may  be  lo- 
cated by  finding  those  of  its 
points.  Let  AB  (Fig.  192) 
represent  an  object  in  front  of 
the  plane  mirror  MN.  Draw 
perpendiculars  from  A  and 
B  to  the  mirror  and  produce 
them  until  their  length  is 
doubled.  A  Bf  is  the  image  of 


N 


Fig.  192 


AB.     It  is  virtual,  erect,  and  of  the  same  size  as  the  object. 

1  In  the  two  right  triangles  ABK  and  A'BK  the  angle  KAB  =  ABF  = 
FBD  =  BA'K\  that  is,  angle  KAB  =  angle  BA'K.  Hence  the  two  right 
triangles  are  equiangular ;  and  since  they  have  one  side  in  common,  they 
are  equal  and  AK  =  A'K.  The  same  result  is  reached  by  the  triangles 
A'CK;  the  point  A'  is  therefore  common  to  the  two  rays. 


REFLECTION  OF  LIGHT 


191 


239.  Path  of  the  Rays  to  the  Eye.  —  Let  A'Bf  (Fig.  193) 
be  the  image  of  AB  in  the  plane  mirror  MN,  and  E  the 
position  of  the  eye  of  an  observer.  To  find  the  path  of 
the  rays  which  enter  the 
eye  at  E,  draw  straight 
lines  from  Af  and  B'  to 
E.  The  intersections  (7 
and  D  of  these  lines  with 
MN&YG  the  points  of  in- 
cidence of  the  rays  from 
A  and  B  which  are  re- 
flected to  the  eye  at  E. 
In  the  same  way  we  may 
trace  the  path  of  the 


Fig.  193 


rays  for  any  other  position  of  the  eye.  Thus  we  see  that 
while  the  image  does  not  change,  the  rays  which  form  it 
for  one  observer  are  not  those  which  form  it  for  another. 

X* 

240.  Uses  of  the  Plane  Mirror.  —  The  employment  of  the 
plane  mirror  as  a  "  looking  glass  "  dates  from  a  period  of 
great  antiquity.  The  process  of  covering  a  glass  surface  with 
an  amalgam  of  tin  and  mercury  came  into  use  in  Venice 
about  three  centuries  ago.  The  process  of  covering  glass 
with  a  film  of  silver  was  invented  during  the  last  century. 

The  fact  that  the  image  in  a  plane  mirror  is  virtual  has 
been  used  to  produce  many  optical  illusions,  such  as  the 
stage  ghost,  the  magic  cabinet,  the  decapitated  head,  etc. 
To  produce  the  illusion  of  a  ghost,  a  large  sheet  of  unsil- 
vered  plate  glass,  with  its  edges  hidden  by  curtains,  is  so 
placed  that  the  audience  have  to  look  obliquely  through  it 
to  see  the  actors  on  the  stage.  Other  actors,  hidden  from 
direct  view,  and  strongly  illuminated,  are  seen  by  re- 
flection in  the  glass  as  ghostly  images  on  the  stage. 


192 


LIGHT 


241.    Multiple  Reflection.  —  Place  two  mirrors  so  that  their  re- 
flecting surfaces  form  an  angle  (Fig.  194).     If  a  lighted  candle  be 

placed  between  them,  several  images 
may  be  seen  in  the  mirrors;  three 
when  they  are  at  right  angles,  more 
when  the  angle  is  less  than  a  right 
angle.  When  the  mirrors  are  parallel, 
all  the  images  are  in  a  straight  line 
Fig.  1 94  perpendicular  to  the  mirrors. 

The  image  in  one  mirror  serves  as  an  object  for  the 
second  mirror,  and  the  image  in  the  second  becomes  in 
turn  an  object  for  the  first  mirror.  In  Fig.  195  the  two 
mirrors  are  at  right  angles.  0'  is  the  image  of  0  in  AB, 
and  is  found  as  in  §  238.  0'"  is  the  image  of  0'  in  A  C, 
and  is  found  by  the  line  O'O'"  drawn  perpendicular  to 
AC  produced.  0"  is  the  image  of  0  in  AC,  and  since  the 
mirrors  are  at  right  angles,  O'n  is  also  the  image  of  0"  in 
AB.  Onf  is  situated  behind 
the  plane  of  both  mirrors, 
and  no  images  of  it  can  be 
formed.  All  the  images  are  o'-'---' 
situated  in  the  circumfer- 
ence of  the  circle  whose  cen- 


Fig.  195 


ter  is  A  and  radius  A  0.  If 
E  is  the  position  of  the  eye, 
then  Of  and  0"  are  each  seen 
by  one  reflection,  and  Ollf 
by  two  reflections,  and  for 
this  reason  it  is  less  bright. 
To  trace  the  path  of  a  ray  for  the  image  O"1,  draw  0"'E, 
cutting  AB  at  6,  and  from  the  intersection  b  draw  bO", 
cutting  AC  at  a.  Join  a  0 ;  the  path  of  the  ray  is  OabE. 
It  is  interesting  to  find  the  images  when  the  mirrors 
are  at  various  angles. 


REFLECTION   OF  LIGHT 


193 


242.  Applications.  —  The  double  image  of  a  bright  star  and  the 
several  images  of  a  gas  jet  in  a  thick  mirror  (Fig.  196)  are  examples 
of  multiple    reflection,   the   front    surface   of    the   mirror  and   the 
metallic    surface   at    the  back 

serving  as  parallel  reflectors. 
Geometrically  the  number  of 
images  is  infinite;  but  on  ac- 
count of  their  faintness  only  a 
limited  number  is  visible.  The 
kaleidoscope,  a  toy  invented  by 
Sir  David  Brewster,  is  an  inter- 
esting application  of  the  same 
principle.  It  consists  of  a  tube 
containing  three  mirrors  extend- 
ing its  entire  length,  the  angle 
between  any  two  of  them  being 
60°.  One  end  of  the  tube  is 
closed  by  ground  glass,  and  the 
other  by  a  cap  with  a  round 
hole  in  it.  Pieces  of  colored 
glass  are  placed  loosely  between 
the  ground  glass  and  a  plate  of 
clear  glass  parallel  to  it.  On 
looking  through  the  whole  at  any  source  of  light,  multiple  images  of 
these  pieces  of  glass  are  seen,  symmetrically  arranged  around  the 
center,  and  forming  beautiful  figures,  which  vary  in  pattern  with 
every  change  in  the  position  of  the  pieces  of  glass. 

243.  Spherical  Mirrors.  —  A  mirror  is  spherical  when  its 
reflecting  surface  is  a  portion  of  the  surface  of  a  sphere. 
If  the  inner  surface  is  polished  for  reflection,  the  mirror 
is    concave;  if   the  outer  surface,  it   is  convex.     Only   a 

small  portion  of  a  spherical  sur- 
face is  used  as  a  mirror.  In  Fig. 
197  the  center  C  of  the  mirror  MN 
is  the  center  of  curvature  of  the 
sphere  of  which  the  reflecting 
Fig.  197  surface  is  a  part.  The  middle 


Fig.  196 


194 


LIGHT 


point  A  of  the  reflecting  surface  MN  is  the  pole  or  vertex 
of  the  mirror,  and  the  straight  line  AB  passing  through 
the  center  of  curvature  C  and  the  pole  A  of  the  mirror  is 
Us  principal  axis.  Any  other  straight  line  through  the 
center  and  intersecting  the  mirror  is  a  secondary  axis. 
The  figures  of  spherical  mirrors  in  this  chapter  are  sections 
of  a  sphere  made  by  passing  a  plane  through  the  principal 


axis. 


Fig.  198 


Fig.  199 


The  difference  between  a  plane  mirror  and  a  spherical 
one  is  that  the  normals  to  a  plane  mirror  are  all  parallel 
lines,  while  those  of  a  spherical  mirror  are  the  radii  of  the 
surface,  and  all  pass  through  the  center  of  curvature. 

244.  Principal  Focus  of  Spherical  Mirrors.  —  A  focus  is  the 
point  common  to  the  paths  of  all  the  reflected  rays  of 
a  pencil  of  light.  It  is  a  real  focus  if  the  rays  of  light 
actually  pass  through  the  point,  Fig.  198,  and  virtual  if 
they  only  appear  to  do  so  (Fig.  199). 

Let  the  rays  of  the  sun  fall  on  a  concave  spherical  mirror.  Hold 
a  graduated  ruler  in  the  position  of  its  principal  axis,  and  slide  along 
it  a  small  strip  of  cardboard.  Find  the  point  where  the  image  of  the 
sun  is  smallest.  This  will  mark  the  principal  focus,  and  it  is  a  real 
one.  If  a  convex  spherical  mirror  be  used,  the  light  will  be  reflected 
as  a  broad  pencil  diverging  from  a  point  back  of  the  mirror.  The 
focus  is  then  a  virtual  one. 


REFLECTION   OF  LIGHT  195 

If  a  pencil  of  parallel  rays  falls  on  a  concave  spherical 
mirror,  parallel  to  its  principal  axis,  the  point  to  which 
the  rays  converge  after  reflection  is  called  the  principal 
focus  of  the  mirror.  In  the  case  of  a  convex  spherical 
mirror,  the  principal  focus  is  the  point  on  the  axis  behind 
the  mirror  from  which  the  reflected 
rays  diverge.  The  distance  of  the 
principal  focus  from  the  mirror  is 
its  principal  focal  length. 


245.  The  Position  of  the  Principal 
Focus.  — Let  MN  (Fig.  200)  be  a 
concave  mirror  whose  center  is  at 

O  and  principal  axis  is  AB.  Let  ED  be  a  ray  parallel  to 
BA.  Then  CD  is  the  normal  at  D ;  and  CDF,  the  angle 
of  reflection,  must  equal  EDO,  the  angle  of  incidence. 
Since  the  ray  BA  is  normal  to  the  mirror,  it  will  be  re- 
flected back  along  AB.  The  reflected  rays  DF  and  AB 
have  a  common  point  F,  which  is  the  principal  focus. 
The  triangle  CFD  is  isosceles  with  the  sides  OF  and  FD 
equal.  (Why?)  But  when  the  point  D  is  near  A,  FD 
is  equal  to  FA ;  F  is  therefore  the  middle  point  of  the 

radius  CA.  Other  rays  par- 
allel to  BA  will  pass  after 
reflection  nearly  through  F. 
Hence,  the  principal  focus  of 
a  concave  spherical  mirror  is 
~"c real  and  is  halfway  between 
the  center  of  curvature  and 
the  vertex. 

Fig.  201  Let  MN  (Fig.  201)  be  a 

convex  spherical  mirror.  ED  and  BA  are  rays  parallel  to 
the  principal  axis.  When  produced  back  of  the  mirror, 


196  LIGHT 

after  reflection,  their  common  point  F  is  back  of  the 
mirror  and  halfway  between  A  and  C.  (Why?)  Hence, 
the  principal  focus  of  a  convex  spherical  mirror  is  virtual 
and  halfway  between  the  center  of  curvature  and  the  mirror. 

246.    Conjugate   Foci   of   Mirrors.  —  When    a   diverging 
pencil  of  light  ABD  (Fig.   202)  falls  on  the  spherical 


M 


Fig.  202 

mirror  MN,  it  is  focused  after  reflection  at  a  point  F 
on  the  axis  AB  which  passes  through  the  radiant  point  or 
source  of  light;  after  reflection  the  rays  diverge  from  this 


Fig.  203 

focus  F  as  a  new  radiant  point.  When  rays  diverging 
from  one  point  converge  to  another,  the  two  points  are 
called  conjugate  foci. 

In  Fig.  203,  the  rays  BA  and  BD  diverge  from  B  as  the 
radiant  point;  after  reflection  they  diverge  as  if  they 
came  from  B'  behind  the  reflecting  surface ;  B'  is  a 
virtual  focus  and  B  and  B1  are  conjugate  foci. 


REFLECTION   OF  LIGHT  197 

In  the  first  case  the  source  of  light  is  farther  from  the 
mirror  than  the  center  of  curvature,  and  the  focus  is  real ; 
in  the  second  case  it  is  nearer  the  mirror  than  the  principal 
focus,  and  the  focus  is  virtual.1 

247.  Images  in  Spherical  Mirrors.  —  In  a  darkened  room 
support  on  the  table  a  concave  spherical  mirror,  a  lamp, 
and  a  small  white  screen.  Place  the  lamp  anywhere 
beyond  the  focus,  and 
move  the  screen  until  a 
clear  image  of  the  flame 
.is  formed  on  it  (Fig.  204). 
Notice  the  size  and  posi- 
tion of  the  image,  and 
whether  it  is  erect  or  in- 
verted. When  the  lamp  is  Fig*  20' 
between  the  focus  and  the  mirror,  an  image  of  it  cannot  be 
obtained  on  the  screen,  but  it  can  be  seen  by  looking  into 
the  mirror.  The  same  is  true  for  the  convex  mirror, 
whatever  be  the  position  of  the  lamp  ;  in  these  last  cases 
the  image  is  a  virtual  one. 

The  experiment  shows  the  relative  positions  of  the 
object  and  its  image  for  a  concave  mirror,  all  depend- 
ing on  the  position  of  the  object  with  respect  to  the 
mirror.  If  these  positions  are  carefully  noted  it  will  be 
seen  that  there  are  six  distinct  cases  as  follows  : 


i  In  Fig.  202,  CD  bisects  the  angle  SDH.     Hence,  -        =  If  D 

B  D     B  C 

is  close  to  A,  we  may,  without  sensible  error,  place  BD  =  BA  and 
B'D  =  B'A.     Put  BA  =  p,  B'A  =  q,   CA  =  r  =  2/.     Then  BC  =  p-r, 

B'C  =  r  —  q,  and  2  =£JH-T  from  which  -  +  -  =  -  =  -•     By  measuring  p 
q     r-q  p     q     r     f 

and  g,  we  may  compute  r  and  /.     For  the  convex  mirror,  q  and  r  are 
negative. 


198 


LIGHT 


First.  —  When  the  object  (ABy  Fig.  205)  is  at  a  finite 
distance  beyond  the  center  of  curvature,  the  image  is  real, 

inverted,  smaller 
than  the  object, 
and  between  the 
center  of  curva- 
ture and  the  prin- 
cipal focus. 
Fig.  20;  /N  Second.— When 

the  object  is  between  the  center  and  the  principal  focus, 

the  image  is  real,  inverted,  larger  than  the  object,  and 

is    beyond    the     center 

(Fig.     206).      This    is 

the    converse    of     case 

one. 

Third.  —  When  a 
small  object  is  at  the 
center  of  curvature,  the 
image  is  real,  inverted, 
of  the  same  size  as  the 
object,  and  at  the  center 
of  curvature  (Fig.  207) .  Fig'  206 

Fourth.  —  When  the  object  is  at  the  principal  focus, 
the  rays  are  reflected  parallel  to  the  principal  axis,  and 
no  image  is  formed  (Fig.  208). 


4.    ^  ^ 


Fig.  207 


Fig.  208 


REFLECTION  OF  LIGHT 


199 


Fifth.  —  When  the  object  is  between  the  principal 
focus  and  the  mirror,  the  image  is  virtual,  erect,  and 
larger  than  the  object  (Fig.  •£. 

209).      . 

Sixth.  —  When  the  mirror 
is  convex,  the  image  is  al- 
ways virtual,  erect,  and 
smaller  than  the  object  (Fig. 
210). 


N 


248.  Construction  for  Images. 

—  To  find  images  in  spherical 
mirrors  by  geometrical  construction,  it  is  only  necessary  to 
find  conjugate  focal  points.  To  do  this  trace  two  rays  for 
each  point  for  the  object,  one  along  the  secondary  axis 
through  it,  and  the  other  parallel  to  the  principal  axis. 
The  first  ray  is  reflected  back  on  itself,  and  the  second 
through  the  principal  focus.  The  intersection  of  the  two 

reflected  rays  from  the 
same  point  of  the  object 
locates  the  image  of 
that  point. 

For  instance:  In  Fig. 
205,  AC  is  the  path  of 
both  the  incident  and 
the  reflected  ray,  while 
the  ray  AD  is  reflected 
through  the  principal 
Fig'210  focus  F.  Their  intersec- 

tion is  at  a.  The  rays  BO  and  BE  are  reflected  similarly 
through  b.  Hence,  ab  is  the  image  of  AB.  In  Fig.  209, 
the  ray  AC  along  the  secondary  axis,  and  AD  reflected 
back  through  F  as  DF,  must  be  produced  to  meet  back  of 


200 


LIGHT 


the  mirror  at  the  virtual  focus  a.     A  and  a  are  conjugate 
foci ;  also  B  and  5,  and  ab  is  a  virtual  image. 

For  the  convex  mirror  (Fig.  210)  the  construction  is 
the  same.  From  the  point  A  draw  AC  along  the,  normal 
or  secondary  axis,  and  AD  parallel  to  the  'principal  axis. 
The  latter  is  reflected  so  that  its  direction  passes  through 
F.  The  intersection  of  these  two  lines  is  at  a.  The 
image  ab  is  virtual  and  erect. 

249.  Spherical  Aberration  in  Mirrors.  —  Bend  a  strip  of 
bright  tin  into  as  true  a  semicircle  as  possible  and  fasten  it 

to  a  vertical  board 
as  in  Fig.  211. 
At  right  angles  to 
the  board  at  one 
end  place  a  vertical 
sheet  of  cardboard 
containing  three 
parallel  slots.  Send 


Fig.  211 


a  strong  beam  of 
light  through  each 
of  these  slots;  the  three  beams  will  be  reflected  by  the  curved 
tin  through  different  points,  the  beam  nearest  the  straight 
rim  of  the  mirror  crossing  the  axis  nearest  the  mirror. 

The  experiment  shows  that  rays  .^ 

incident  near,  the  margin  of  a  spher- 
ical mirror  cross  the  axis  after  reflec- 
tion between  the  principal  focus  and 
the  mirror.  This  spreading  out  of 
the  focus  is  known  as  spherical  aber- 
ration by  reflection.  It  causes  a  lack 
of  sharpness'in  the  outline  of  image's 
formed  by  spherical  mirrors.  It  is  Fig.  212 


/ 

/l 

1 

/v 

'  f 

Pi 

/ 

\^/ 

\F                    c 

v> 

v  V 

\ 

\         ; 

NX                * 

REFLECTION  OF  LIGHT 


201 


Fig.  213 


reduced  by  decreasing  the  aperture  of  the  mirror  by  means 
of  a  diaphragm  to  cut  off  marginal  rays,  or  by  decreasing 
the  curvature  of 
the  mirror  from 
the  vertex  out- 
ward. The  result 
then  is  a  parabol- 
ic mirror  (Fig. 
212),  which  finds 
use  in  search- 
lights, light- 
houses, head- 
lights of  loco- 
motives, and  in 
reflecting  tele- 
scopes. 

250.   Caustics  by  Reflection.  —  Use  the  tin  reflector  of  the 
last  experiment  as  shown  in  Fig.   213.     The  light  from 

a  candle  or  a  lamp  is  focused 
on  a  curved  line. 

The  curve  formed  by  the 
rays  reflected  from  a  spherical 
mirror  is  called  the  caustic  by 
reflection.  It  may  be  seen  by 
letting  sunlight  fall  on  a  tin 
milk  pail  partly  full  of  milk, 
or  on  a  plain  gold  ring  on  a 
white  surface.  The  caustic 
curve  may  be  constructed  by 
drawing  a  series  of  parallel 
rays  incident  on  a  concave 
mirror  of  large  aperture  (Fig. 
214),  and  tracing  the  reflected 


202  LIGHT 

rays  by  means  of  tta  law  of  reflection,  that  is,  making  in 
each  case  the  angle  m  reflection  equal  to  the  angle  of  in- 
cidence. The  caustic  is  the  curved  line  to  which  all  of 
the  reflected  rays  are  tangent.  It  should  be  noticed  that 
in  the  case  of  a  concave  spherical  mirror  the  caustic  is  a 
surface. 

Problems 

1.  Two  mirrors  are  placed  at  an  angle  of  45°.     Find  graphically 
how  many  images  there  are  of  a  point  situated  between  the  mirrors. 

2.  Show  by  a  diagram  that  a  person  can  see  his  whole  length  in 
a  mirror  of  half  his  height. 

3.  Which  will  give  the  stronger  illumination  of  your  book,  a  16 
candle  power  light  at  a  distance  of  2  ft.,  or  a  32  candle  power  light  at 
a  distance  of  4  ft.  ? 

4.  An  object  placed  8  ft.  in  front  of  a  concave  spherical  mirror 
gave  an  image  2  ft.  from  the  mirror.     What  was  the  principal  focal 
length  of  the  mirror? 

5.  An  object  is  placed  20  cm.  in  front  of  a  concave  mirror  having 
a  principal  focal  length  of  30  cm.     How  far  back  of  the  mirror  is  the 
image  ? 

6.  'Where  must  an  object  be  placed  in  front  of  a  concave  spherical 
mirror  to  get  an  image  halfway  between  the  center  of  curvature  and 
the  principal  focus  ? 

7.  An  object  is  placed  20  in.  in  front  of  a  concave  spherical  mir- 
ror of  10  in.  radius.     Find  the  position  of  Ttte  image. 

8.  Find  the  radius  of  curvature  of  a  concave  spherical  mirror 
when  an  object  100  cm.  from  the  mirror  gives  a  real  image  50  cm. 
from,  the  mirror. 

9.  By  making/,  r,  and  q  negative  in  the  formula  for  a  concave 
mirror,  it  applies  to  a  convex  mirror.     If  the  radius  of  curvature  of  a 
convex  spherical  mirror  is  20  in.,  what  is  the  position  of  the  image 
when  the  object  is  5  ft.  from  the  mirror? 

10.  If  a  plane  mirror  is  moved  parallel  to  itself  directly  away  from 
an  object  in  front  of  it,  how  much  faster  does  the  image  move  than 
the  mirror? 


REFRACTION  OF  LIGHT 


203 


IV.    REFRACTION  OF  LIGHT 

251.  Refraction.  —  Fasten  a  paper  protractor  scale  centrally  on 
one  face  of  a  retarigular  battery  jar  (Fig.  215),  and  fill  the  jar  with 
water  to  the  horizontal  diameter 
of  the  scale.  Place  a  slotted  card- 
board over  the  top.  With  a  plane 
mirror  reflect  a  beam  of  light 
through  the  slit  into  the  jar,  at 
such  an  angle  that  the  beam  is 
incident  on  the  water  exactly  back 
of  the  center  of  the  scale.  The 
path  of  this  ribbon  of  light 
may  be  traced;  its  direction  is 
changed  at  the  surface  of  the 
water. 

The  change  in  the  course 

of   light  in  passing  from   one  transparent  medium  into 
another  is  called  refraction. 

Place  a  coin  at  the  bottom  of  an  empty  cup  standing  on  a  table, 
and  let  an  observer  move  back  until  the  coin  just  passes  out  of  sight 

...  below  the  edge  of  the  cup ;  now 
pour  water  into  the  cup,  and  the 
coin  will  come  into  view  (Fig.  216). 

The  changes  in  the  apparent  depth 
of  a  pond  or  a  stream,  as  the  observer 
moves  away  from  it,  are  caused  by 
refraction.  The  broken  appearance 
of  a  straight  pole  thrust  obliquely 
into  water  is  accounted  for  by  the 
change  in  direction  which  the  rays 
coming  from  the  part  under  water 
Fig.  2 1 6  suffer  as  they  emerge  into  the  air. 

252.  Cause  of  Refraction.  —  Foucault  in  France  and 
Michelson  in  America  have  measured  the  velocity  of  light 
in  water,  and  have  found  that  it  is  only  three  fourths  as 


204 


LIGHT 


M 


great  as  in  air.  The  velocity  of  light  in  all  transparent 
liquids  and  solids  is  less  than  in  air,  while  the  velocity  in 
air  is  practically  the  same  as  in  a  vacuum. 

If  now  a  beam  of  light  is  incident  obliquely  on  the  sur- 
face MN  of  water  (Fig.  217),  all  parts  of  a  wave  do  not 

enter  the  water  at  the 
same  time.  Let  the  par- 
allel lines  perpendicular 
to  AB  represent  short 
portions  of  plane  waves. 
Then  one  part  of  a  wave, 
as  /,  will  reach  the  water 
before  the  other  part,  as 
e,  and  will  travel  less 
rapidly  in  the  water  than 
in  the  air.  The  result 
is  that  each  wave  is 
swung  around,  that  is, 
the  direction  of  propaga- 
tion BG,  which  is  .perpendicular  to  the  wave  fronts,  is 
change^ ;  in  other  words,  the  beam  is  refracted.  The 
refraction  of  light  is  therefore  owing  to  its  change  in 
velocity  in  passing  from  one 
transparent  medium  to  another. 


253.   The  Index  of  Refraction. 

—  Let  a  beam  of  light  pass 
obliquely  from  air  to  water  or 
glass,  and  let  AB  (Fig.  218) 
be  the  incident  wave  front. 
From  A  as  a  center  and  with 


Fig.  217 


Fig.  218 


a  radius  AD  equal  to  the  distance  the  light  travels  in  the 
second  medium  while  it  is  going  from  B  to  C  in  air,  draw 


REFRACTION  OF  LIGHT  205 

the  dotted  arc.  This  limits  the  distance  to  which  the  dis- 
turbance spreads  in  the  second  medium.  Then  from  C 
draw  CD  tangent  to  this  arc  and  draw  AD  to  the  point  of 
tangency.  CD  is  the  new  wave  front. 

The  distances  BO  and  AD  are-traversed  by  the  light  in 
the  same  time.  They  are  therefore  proportional  to  the 
velocities  of  light  in  the  two  media.  Then 

the  index  of  refraction      A- 

_  the  velocity  of  light  in  air  _  v  l 

the  velocity  of  light  in  the  second  medium      v' 

The  angle  NCB  is  the  angle  of  incidence.  It  is  equal 
to  the  angle  BAG  between  the  incident  wave  front  and 
the  surface  of  separation  of  the  two  media.  The  angle  of 
refraction  is  the  angle  N7  AD.>  It  is  equal  to  the  angle 
ACD  between  the  wave  front  in  the  second  medium  and 
the  surface  of  separation.  The  angle  at  (7,  between  the 
direction  of  the  incident  ray  and  the  refracted  ray,  is  the 
angle  of  deviation. 

The  following  are  the  indices  of  refraction  for  a  few 
substances  : 

Water          .     .     .  1.33  Crown  glass      .     .     .   1.51 

Alcohol        .     .     .1.36  Flint  glass    .     1.54  to  1.71 

Carbon  bisulphide  1.64  Diamond       ....  2.47 

.  1  The  older  mathematical  definition  of  the  index  of  refraction  is  the 
ratio  of  the  sine  of  the  angle  of  incidence  to  the  sine  of  the  angle  of  re- 
fraction. Now  the  sine  of  an  angle  in  a  right  triangle  is  the  quotient  of 
the  side  opposite  by  the  hypotenuse.  Thus,  the  sine  of  angle  SAC  is 

— ,  and  the  sine  of  ACD  is  "==-  •  Dividing  one  by  the  other,  the  common 
AC  AC 

TiC      11 

term  A  C  cancels  out,  and  the  index  of  refraction  equals  —  =  -. ,  as  before. 

AD      v 

The  two  definitions  are  therefore  equivalent  to  each  other.  For  the  con- 
struction to  find  the  refracted  ray,  see  the  Appendix. 


206 


LIGHT 


254.  Laws  of  Refraction.  —  The  following  laws,  which 
summarize  the  facts  relative   to   single   refraction,  were 
discovered  by  Snell,  a  Dutch  physicist,  in  1621: 

I.  When  a  pencil  of  light  passes  obliquely  from  a  less 
highly  •  to  a  more  highly  refractive  medium,  it_  is  bent 
toward  the  normal ;  when  it  passes  in  the  reverse  direc- 
tion, it  is  bent  from  the  normal. 

II.  Whatever  the  angle  of  incidence,  the  index  of  re- 
fraction is  a  constant  for  the  same  two  media. 

III.  The  planes  of  the  angles  of  incidence  and  refrac- 
tion coincide. 

255.  Refraction  through  Plate    Glass.  —  Draw   a  heavy 
black  line  on  a  sheet  of  paper,  and  place  over  it  a  thick 

plate  of  glass,  cover- 
ing a  part  of  the 
line.  Look  obliquely 
through  the  glass;  the 
line  will  appear  bro- 
ken at  the  edge  of 
4ihe  plate,  the  part 
under  the  glass  ap- 
pearing laterally  dis- 
placed (Fig.  219). 


Fig.  219 


Fig.  220 


To  explain  this,  let  J£ZV(Fig.  220)  represent  a  thick 
plate  of  glass,  and  AB  a  ray  of  light  incident  obliquely 
upon  it.  If  the  path  of  the  ray  be  determined,  the  emer- 
gent ray  will  be  parallel  to  the  incident  ray.  Hence,  the 
apparent  position  of  an  object  viewed  through  a  plate  is 
at  one  side  of  its  true  position. 

256.  A  Prism.  —  Let  ABQ  (Fig.  221)  represent  a  section 
of  a  glass  prism  made  by  a  plane  perpendicular  to  the  re- 


v 


REFRACTION   OF  LIGHT 


207 


fracting  edge  A.  Also,  let  LI  be  a  ray  incident  on  the 
face  BA.  This  ray  will  be  refracted  along  IE,  and  enter- 
ing the  air  at  the  point  E  will 
be  refracted  again,  taking  the  di- 
rection EO. 

Reflect  across  the  table  a  strong  beam 
of  light  and  intercept  it  with  a  sheet  of 
green  glass.  Let  this  ribbon  of  green 
light  be  incident  on  a  prism  of  small  re- 
fracting angle  in  such  a  manner  that 
only  part  of  the  beam  passes  through  the 


Fig.  221 


prism.  Two  lines  of  light  may  be  traced  through  the  dust  of  the  room 
or  by  means  of  smoke.  By  turning  the  prism  about  its  axis,  the 
angle  between  these  lines  of  light  can  be  varied  in  size.  It  is  the 
angle  of  deviation,  represented  by  the  angle  D  in  the  figure.  The 
angle  of  deviation  is  least  when  the  angles  of  incidence  and  emer- 
gence are  equal ;  this  occurs  when  the  path  of  the  ray  through  the 
prism  is  equally  inclined  to  the  two  faces. 

257.  Atmospheric  Refraction.  —  Light  coming  to  the  eye 

from  any  heavenly  body,  as  a  star,  unless  it  is  directly 

s^  overhead,    is    gradually 

x\  bent  as  it  passes  through 

the  air  on  account  of  the 
increasing  density  of 

^Horizon  J 

!#^w\  ^ne  atmosphere  near  the 

earth's  surface.  Thus, 
if  8  in  Fig.  222  is  the 
real  position  of  a  star,  its 
apparent  position  will  be 
AS"  to  an  observer  at  E. 

Such  an  object  appears  higher  above  the  horizon  than  its 
real  altitude.  The  sun  rises  earlier  on  account  of  atmos- 
pheric refraction  than  it  otherwise  would,  and  for  the 
same  reason  it  sets  later.  Twilight,  the  mirage  of  the 


208 


LIGHT 


Fig.  223 


desert,  and  the  looming  of  distant  objects  are  phenomena 
of  atmospheric  refraction. 

258.  Total  Internal  Reflection.  —  Take  the  apparatus  of  §  251 
and  place  the  cardboard  against  the  end  of  the  jar  so  that  the  slit  is 

near  the  bottom  (Fig.  223).  Re- 
flect a  strong  beam  of  light  up 
through  the  water  and  incident  on 
its  under  surface  just  back  of  the 
protractor  scale.  Adjust  the  slit 
so  that  the  beam  shall  be  incident 
at  an  angle  a  little  greater  than 
50°.  It  will  be  reflected  back  into 
the  water  as  from  a  plane  mirror. 

As  the  angle  of  refraction 
is  always  greater  than  the 
angle  of  incidence  when  the  light  passes  from  water  into 
air,  it  is  evident  that  there  is  an  incident  angle  of  such  a 
value  that  the  corresponding  angle  of  refraction  is  90°, 
that  is,  the  refracted  light  is  parallel  to  the  surface.  If 
the  angle  of  incidence  is  still  further  increased,  the  light 
no  longer  passes  out  into  the  air,  but  suffers  total  internal 
reflection. 

259.  The  Critical  Angle.  —  The  critical  angle  is  the  angle 
of  incidence  corresponding  to  an  angle  of  refraction  of  90°. 
This  angle  varies  with  the  index  of  refraction  of  the  "sub- 
stance.    It  is  about  49°  for  water,  42°  for  crown  glass,  38° 
for  flint  glass,  and  24°  for  diamond. 

Of  all  the  rays  diverging  from  a 
point  at  the  bottom  of  a  pond  and 
incident  on  the  surface,  only  those 
within  a  cone  whose  semi-angle  is 
49°  pass  into  the  air.  All  those' in- 
cident at  a  larger  angle  undergo 
total  internal  reflection  (Fig.  224).  Fig.  224 


PROBLEMS 


209 


Hence,  an  observer  under  water  sees  all  objects  outside 
as  if  they  were  crowded  into  this  cone ;  beyond  this 
he  sees  by  reflection  objects  on  the 
bottom  of  the  pond. 

Total  reflection  in  glass  is  shown 
by  means  of  a  prism  whose  cross 
section  is  a  right-angled  isosceles 
triangle  (Fig.  225).  A  ray  inci- 
dent normally  on  either  face  about 
the  right  angle  enters  the  prism 
without  refraction,  and  is  incident 
on  the  hypotenuse  at  an  angle  of  45°,  which  is  greater 
than  the  critical  angle.  The  ray  therefore  suffers  total  in- 
ternal reflection  and 
leaves  the  prism  at 
right  angles  to  the 
incident  ray.  A  sim- 
ilar prism  is  some- 
times used  in  a 

Fig.  226  .        . 

projecting      lantern 

for  making  the  image  erect  (Fig.  226).     It  would  other- 
wise be  inverted  with  respect  to  the  object. 

Problems 

1.  Where  would  you  place  a  lamp  in  front  of  a  concave  spherical 
mirror  to  get  an  image  of  the  lamp  on  the  wall  larger  than  the  lamp 
itself  ?    Illustrate  by  a  figure. 

2.  Why  does  the  full  moon  as  it  rises  appear  to  be  oblong  with 
the  major  axis  horizontal? 

3.  What  deviation  is  produced  by  reflection  from  a  plane  mirror 
when  the  angle  of  incidence  is  60°? 

4.  Paste  diamonds  are  made  of  flint  glass  and   have   about   the 
same  index  of  refraction  as  carbon  bisulphide.     A  diamond  is  visible 
in  carbon  bisulphide  and  the  paste  diamond  is  not.     Explain. 


210 


LIGHT 


5.  A  bottle  filled  with  pounded  glass  is  opaque,  and  is  translucent 
when  spirits  of  turpentine  are  added.     Explain. 

6.  Show  by  a  diagram  that  objects  viewed  obliquely  through  a 
plate  glass  window  are  not  seen  exactly  in  their  true  position. 

7.  Show  by  a  diagram  the  effect  of  a  hollow  prism  filled  with  air 
and  submerged  in  water  on  a  beam  of  light  passing  through  the  water 
and  incident  on  the  prism. 

8.  The  index  of  refraction  for  water  is  f .    If  the  velocity  of  light 
in  air  is  186,000  mi.  per  second,  what  is  it  in  water? 

9.  If  the  index  of  refraction  for  crown  glass  is  f,  and  for  water  is 
f ,  compare  the  speed  of  light  in  crown  glass  with  that  in  water.    What 
simple  fraction  represents  the  relative  speed  ? 

10.   Will  a  pencil  of  light  passing  obliquely  from  water  into  flint 
glass  be  bent  toward  or  away  from  the  perpendicular  in  the  glass? 

V.    LENSES 

260.    Kinds  of  Lenses.  —  A  lens  is  a  portion  of  a  trans- 
parent substance  bounded  by  two  surfaces,  one  or   both 


Fig.  227 

being  curved.     The  curved  surfaces  are  usually  spherical 
(Fig.  227).     Lenses  are  classified  as  follows: 


LENSES 


211 


1.  Double-convex,  —  both  surfaces  convex     .     . 

2.  Plano-convex,  —  one     surface     convex,     one 

plane 

3.  Concavo-convex,  —  one  surface  convex,  one 

concave    

4.  Double-concave,  —  both  surfaces  concave 

5.  Plano-concave,  —  one    surface   concave,    one 

plane • . 

6.  Convexo-concave,  —  one  surface  concave,  one 

convex    


Converging  lenses, 

thicker  at  the  middle 

than  at  the  edges. 


Diverging  lenses, 

thinner  at  the  middle 

than  at  the  edges. 


The  concavo-convex  and  the  convexo-concave  lenses  are 
frequently  called  meniscus  lenses.  The  double-convex 
lens  may  be  regarded  as  the  type  of  the  converging  class 
of  lenses,  and  the  double-concave  lens  of  the  diverging 

class. 

261.  Definition  of  Terms  relating  to  Lenses.  —  The  centers 
of  the  spherical  surfaces  bounding  a  lens  are  the  centers  of 
curvature.  The  optical  center  is  a  point  such  that  any  ray 
passing  through  it  and 
the  lens  suffers  no 
change  of  direction.  In 
lenses  whose  surfaces 
are  of  equal  curvature, 
the  optical  center  is  their 
center  of  volume,,  as  0, 
in  Fig.  228.  In  piano- 
lenses,  the  optical  center  is  the  middle  point  of  the  curved 
face.  The  straight  line,  CO',  through  the  centers  of  cur- 
vature, is  the  principal  axis,  and  any  other  straight  line 
through  the  optical  center  as  EH,  is  a  secondary  axis.  The 
normal  at  any  point  of  the  surface  is  the  radius  of  the 
sphere  drawn  to  that  point;  thus  CD  is  the  normal  to 
the  surface  AnB  at  D. 


22S 


212 


LIGHT 


262.  Tracing  Rays  through  Lenses.  —  A  study  of  Figs. 
229  and  230  shows  that  the  action  of  lenses  on  rays  of 
light  traversing  them  is  similar  to  that  of  prisms,  and 
conforms  to  the  principle  illustrated  in  §  256.  A  ray  is 
always  refracted  towar-d  the  perpendicular  on  entering  a 


denser  medium  (glass),  and  away  from  it  on  entering  a 
medium  of  less  optical  density.  Thus  we  see  that  the 
convex  lens  bends  a  ray^toward  the  principal  axis,  while 
the  concave  lens  (Fig.  230)  bends  it  away  from  this  axis. 


263.  The  Principal  Focus.  —  Hold  a  converging  lens  so  that  the 
rays  of  the  sun  fall  on  it  parallel  to  its  principal  axis.  Beyond  the 
lens  hold  a  sheet  of  white  paper,  moving  it  until  the  round  spot 
of  light  is  smallest  and  brightest.  If  held  steadily,  a  hole  may  be 
burned  through  the  paper.  This  spot  marks  the  principal  focus  of 
the  lens,  and  its  distance  from  the  optical  center  is  the  principal 
focal  length. 


LENSES 


213 


Converging  lenses  are  sometimes  called  burning  glasses 
because  of  their  power  to  focus  the  heat  rays,  as  shown  in 
the  experiment. 

.Figure  231  shows  that  parallel  rays  are  made  to  converge 
toward  the  principal  focus  F  by  a  converging  lens,  and 


the  focus  is  real;  on  the  other  hand,  Fig.  232  illustrates 
the  diverging  effect  of  a  concave  lens  on  parallel  rays ; 
the  focus  F  is  now  virtual  because  the  rays  after  passing 


H 


Fig.  232 

through  the  lens  only  apparently  come  from  F.  In  gen- 
eral, converging  lenses  increase  the  convergence  of  light, 
while  diverging  lenses  decrease  it. 

264.  Conjugate  Foci  of  Lenses.  —  If  a  pencil  of  light  di- 
verges from  a  point  and  is  incident  on  the  lens,  it  is 
focused  at  a.  point  on  the  axis  through  the  radiant  point. 


214  LIGHT 

These  points  are  called  conjugate  foci,  for  the  same  reason 
as  in  mirrors. 

In  Fig.  233  a  pencil  of  rays  BAE  diverges  from  A  and 
is  focused  by  the  lens  at  the  point  H.  It  is  evident  that 
if  the  rays  diverge  from  H,  they  would  be  brought  to  a 
focus  at  A.  Hence  A  and  -ffare  conjugate  foci. 


265.  Images  by  Lenses.  —  Place  in  line  on  the  table  in  a  dark- 
ened room  a  lamp,  a  converging  lens  of  known  focal  length,  and  a 
white  screen.  If,  for  example,  the  focal  length  of  the  lens  is  30  cm., 
place  the  lamp  about  70  cm.  from  it,  or  more  than  twice  the  focal 
length,  and  move  the  screen  until  a  clearly  denned  image  of  the  lamp 
appears  on  it.  This  image  will  be  inverted,  smaller  than  the  object, 
and  situated  betw.een  30  cm.  and  60  cm.  from  the  lens.  By  placing 
the  lamp  successively  at  60  cm.,  50  cm.,  30  cm.,  and  20  cm.,.the  images 
will  differ  in  position  and  size,  and  iii  the  last  case  will  not  be  received 
on  the  screen,  but  may  be  seen  by  looking  through  the  lens  toward 
the  lamp.  If  a  diverging  lens  be  used,  no  image  can  be  received  on 
the  screen  because  they  are  all  virtual. 

The  results  of  such  an  experiment  may  be  summarized 
as  follows : 

I.  When  the  object  is  at  a  finite  distance  from  a  con- 
verging lens,  and  farther  than  twice  the  focal  length,  the 
image  is  real,  inverted,  at  a  distance  from'the  lens  of  more 
than  once  and  less  than  twice  the  focal  length,  and  smaller 
than  the  object  (Fig.  234). 


215 


LENSES 

II.    When  the  object  is  at  a  distance  of  twice  the  focal 
length  from  a  converging  lens,  the  image  is  real,  inverted, 


Fig.  234 

at  the  same  distance  from  the  lens  as  the  object,  and  of 
the  same  size  (Fig.  235). 

III.    When   the   object  is  at  a  distance  from   a   con- 
verging lens  of  less  than  twice  and  more  than  once  its 


Fig.  235 

focal  length,  the  image  is  real,  inverted,  at  a  distance  of 
more  than  twice  the  focal  length,  and  larger  than  the 
object  (Fig.  236). 


Fig.  236 


216 


LIGHT 


IV.    When  the  object  is  at  the  principal  focus  of  a  con 
verging  lens,  no  distinct  image  is  formed  (Fig.  237). 


Fig.  237 

V.  When  the  object  is  between  a  converging  lens  and 
its  principal  focus,  the  image  is  virtual,  erect,  and  en- 
larged (Fig.  238). 


Fig.  238 


VI.    With  a  diverging  lens,  the  image  is  always  virtual, 
erect,  and  smaller  than  the  object  (Fig.  239). 


Fig.  239 


LENSES  217 

266.  Graphic  Construction  of  Images  by  Lenses.  —  The  im- 
age of  an  object  by  a  lens  consists  of  the  images  of  its 
points.     If  the  object  is  represented  by  an  arrow,  it  is 
necessary  to  find  only  the  images  of  its  extremities.     This 
is  readily  done  by  following  two  general  directions : 

First.  Draw  secondary  axes  through  the  ends  of  the 
arrow.  These  represent  rays  that  suffer  no  change  in 
direction  because  they  pass  through  the  optical  center 
(§  261). 

Second.  Through  the  ends  of  the  arrow  draw  rays 
parallel  to  the  principal  axis.  After  leaving  the  lens, 
these  pass  through  the  principal  focus  (§  263). 

The  intersection  of  the  two  refracted  rays  from  each 
extremity  will  be  its  image. 

To  illustrate.  Let  AB  be  the  object  and  MN  the  lens 
(Figs.  234-239).  Rays  along  secondary  axes  through  0 
pass  through  the  lens  without  any  change  in  direction. 
The  rays  AD  and  BH,  parallel  to  the  principal  axis,  are 
refracted  in  the  lens  along  DE  and  HI  respectively,  and 
emerge  from  the  lens  in  a  direction  which  passes  through 
the  principal  focus  F.  The  intersection  of  Aa  with  Ea 
is  the  image  of  A,  and  that  of  Bb  with  Ib  is  the  image  of 
B.  Other  rays  from  A  and  B  also  pass  through  a  and  b 
respectively,  and  therefore  ab  is  the  image  of  AB.  The 
image  is  virtual  when  the  intersection  of  the  refracted 
rays  is  on  the  same  side  of  the  lens  as  the  object.  The 
relative  size  of  object  and  image  is  the  same  as  their  rela- 
tive distance  from  the  lens. 

267.  Spherical  Aberration  in,  Lenses.  —  If  rays  from  any 
point  be  drawn  to   different  parts  of  a  lens,  and  their 
directions  be  determined  after  refraction,  it  will  be  found 
that  those  incident  near  the  edge  of  the  lens  cross  the 


218  LIGHT 

principal  axis,  after  emerging,  nearer  the  lens  than  those 
incident  near  the  middle.  The  principal  focal  length  for 
the  marginal  rays  is  therefore  less  than  for  central  rays. 
This  indefiniteness  of  focus  is  called  spherical  aberration 
by  refraction,  the  effect  of  which  is  to  lessen  the  distinct- 
ness of  images  formed  by  the  lens.  In  practice  a  round 
screen,  called  a  diaphragm,  is  used  to  cut  off  the  marginal 
rays ;  this  renders  the  image  sharper  in  outline,  but  less 
bright.  In  the  large  lenses  used  in  telescopes  the  curva- 
ture of  the  lens  is  made  less  toward  the  edge,  so  that  all 
parallel  rays  are  brought  to  the  same  focus. 


Fig.  240 
268.  Formula  for  Lenses.  — The  triangles  A  OK  and  aOL  in 

4  ~K       KT) 

Fig.  240  are  similar.    Hence,  ££==££.     If  the  lens  is  thin,  a  straight 

aL       LO 

line  connecting  D  and  H  will  pass  very  nearly  through  the  optical 

center  0.     Then  DFO  is  a  triangle  similar  to  aFL,  and  52  =s  ~L; 

aL       Lr 

Since  DO  is  equal  to  AK,  the  first  members  of  the  two  equations 

K '  O      O  F 

above  are  equal  to  each  other,  and  therefore  — —  =  — — .     Put  KO  =  p, 

LO     Lb 

LO-q,  and  OF  =/.     Then  LF  =  q  -/,  and 

P=    f 

q    ?-/' 

Clearing  of  fractions  and  dividing  through  by  pqf,  we  have 

i  =  i  +  -.  .     .      (Equation  32) 

/>--.;? 

By  measuring  p  and  q  we  may  compute  /.     For  diverging  lenses 
/  and  q  are  negative. 


OPTICAL  INSTRUMENTS  219 

Questions  and  Problems 

1.  Given  a  spectacle  lens,  how  will  you  determine  whether  it  is 
converging  or  diverging  ? 

2.  Where  must  the  observer  place  himself  so  as  to  see  his  own 
image  in  a  concave  mirror  ? 

3.  Why  is  the  image  of  yourself  in  the  bowl  of  a  silver  spoon 
distorted? 

4.  What  kind  of  mirrors  and  lenses  always  produce  virtual  images  ? 

5.  If  one  half  of  a  converging  lens  is  covered  by  an  opaque  card, 
what  will  be  the  effect  on  the  real  image  ? 

6.  When  an  object  moves  from  a  great  distance  up  to  twice  the 
focal  length  of  a  converging  lens,  how  far  does  the  image  move?  ; 

x7.  A  candle  is  placed  10  ft.  from  a  white  wall.  Find  the  position 
of  a  converging  lens  that  will  give  an  enlarged  image  of  the  candle  on 
the  wall,  the  focal  length  of  the  lens  being  20  in. 

8.  Why  does  a  mirror  made  of  plate  glass  give  a  better  image  than 
one  made  of  common  window  glass? 

9.  What  is  the  smallest  distance  between  an  object  and  its  real 
image  in  a  converging  lens,*expressed  in  terms  of  the  focal  length? 

10.  The  focal  length  of  a  camera  lens  is  20  cm.     How  far  from 
the  lens  must  the  sensitized  plate  be  placed  when  the  object  is  200  cm. 
from  the  lens  ? 

11.  An  object  5  cm.  long  in  front  of  a  converging  lens  has  an 
image  20  cm.  long  on  a  screen  100  cm.  from  the  lens.     What  is  the 
focal  length  of  the  lens? 

/ 12.    An  object  is  placed  100  cm.  from  a  diverging  lens  whose  focal 
length  is  33£  cm.     What  is  the  distance  of  the  virtual  image? 

VI.     OPTICAL  INSTRUMENTS 

269.  The  Magnifying  Glass,  or  simple  microscope,  is  a 
double-convex  lens,  usually  of  short  focal  length.  The 
object  must  be  placed  nearer  the  lens  than  its  principal 
focus.  The  image  is  then  virtual,  erect,  and  enlarged. 


• 


220 


LIGHT 


If  AJ3  is  the  object  in  Fig.  241,  the  virtual  image  is  ab  ; 
and  if  the  eye  be  placed  near  the  lens  on  the  side  opposite 


M 


the  object  the  virtual  image  will  be  seen  in  the  position  of 
the  intersection  of  the  rays  produced,  as  at  ab. 

270.   The   Compound   Microscope  (Fig.  242)  is  an  instru- 
ment  designed  to  obtain   a   greatly  enlarged   image   of 

very  small  objects.  In  its 
simplest  form  it  consists  of 
a  converging  lens  MN  (Fig. 
243),  called  the  object  glass 
or  objective,  and  another  con- 
verging lens  RS,  called  the 
eyepiece.  The  two  lenses  are 
mounted  in  the  ends  of  the 
tube  of  Fig.  242.  The  ob- 
ject is  placed  on  the  stage 
just  under  the  objective, 
and  a  little  beyond  its  prin- 
cipal focus.  A  real  image 
ab  (Fig.  243)  is  formed 
slightly  nearer  the  eyepiece 
than  its  focal  length.  This 
image  formed  by  the  objective  is  viewed  by  the  eye- 
piece, and  the  latter  gives  an  enlarged  virtual  image. 


242 


OPTICAL  INSTRUMENTS 


221 


(Why  ?)     Both  the  objective  and  the  eyepiece  produce 
magnification. 

271.  The  Astronomical  Telescope.  —  The  system  of  lenses 
in  the  refracting  astronomical  telescope  (Fig.  244)  is  simi- 
lar to  that  of  the  compound  microscope.  Since  it  is  in- 
tended to  view  distant  objects,  the  objective  MN  is  of 


large  aperture  and  long  focal  length.  The  real  image 
given  by  it  is  the  object  for  the  eyepiece,  which  again 
forms  a  virtual  image  .for  the  eye  of  the  observer.  The 
magnification  is  the  ratio  of  the  focal  lengths  of  the  objec- 
tive and  the  eyepiece.  The  objective  must  be  large,  for 


the  purpose  of  collecting  enough  light  to  permit  large 
magnification  of  the  image  without  too  great  loss  in 
brightness. 

Figure  244  shows  that  the  image  in  the  astronomical 
telescope  is  inverted.  In  a  terrestrial  telescope  the  image 
is  made  erect  by  introducing  near  the  eyepiece  two  double- 


222 


LIGHT 


convex  lenses,  in  such  relation  to  each  other  and  to  the 
first  image  that  a  second  real  image  is  formed  like  the 
first,  but  erect. 

272.  Galileo's  Telescope.  —  The  earliest  form  of  telescope 
was  invented  by  Galileo.  It  produces  an  erect  image  by 
the  use  of  a  diverging  lens  for  the  eyepiece  (Fig.  245). 


Fig.  245 

This  lens  is  placed  between  the  objective  and  the  real 
image,  ab,  which  would  be  formed  by  the  objective  if  the 
eyepiece  were  not  interposed.  Its  focus  is  practically  at 
the  image  «J,  and  the  rays  of  light  issue  from  it  slightly 
divergent  for  distant  objects.  The  image  is  therefore  at 
AJB'  instead  of  at  ab,  and  it  is  erect  and  enlarged.  This 
telescope  is  much  shorter  than  the  astronomical  telescope, 
for  the  distance  between  the  lenses  is  the  difference  of 
their  focal  lengths  instead  of  their  sum.  In  the  opera 
glass  two  of  Galileo's  telescopes  are  attached  together 
with  their  axes  parallel. 

273.  The  Projection  Lantern  is  an  apparatus  by  which  a 
greatly  enlarged  image  of  an  object  can  be  projected  on  a 
screen.  The  three  essentials  of  a  projection  lantern  are 
a  strong  light,  a  condenser,  and  an  objective.  The  light 
may  be  the  electric  arc  light,  as  shown  in  Fig.  246,  the 
calcium  light,  or  a  large  oil  burner.  The  condenser  E  is 


OPTICAL  INSTRUMENTS 


223 


composed  of  a  pair  of  converging  lenses;  its  chief  pur- 
pose is  the  collection  of  the  light  on  the  object  by 
refraction,  so  as  to  bring  as  much  as  possible  on  the 
screen.  The  object  AB,  commonly  a  drawing  or  a  photo- 
graph on  Lglass,  is  placed  near  the  condenser  SS,  where 


Fig.  246 

it  is  strongly  illuminated.  The  objective,  MN,  is  a  com- 
bination of  lenses,  acting  as  a  single  lens  to  project  on  the 
screen  a  real,  inverted,  and  enlarged  image  of  the  object. 

274.    The   Photographer's   Camera  consists   of  a  box  L8 
(Fig.  247),  adjustable  in  length,  blackened  inside,  and 
provided    at    one 
end  with  a  lens  or 
a   combination   of 
lenses,    acting    as 
a  single  one,  and 
at  the  other  with          ^ 
a  holder    for   the     ^m'-fl^w 
sensitized    plate.  Fig.  247 

If  by  means- of  •rack  and  pinion  the  lens  L'  be  prop- 
erly focused  for  an  object  in  front  of  it,  an  inverted 
image  will  be  formed  on  the  sensitized  plate  E.  The 
light  acts  on  the  salts  contained  in  the  sensitized  film, 
producing  in  them  a  modification  which,  by  the  processes 


224 


LIGHT 


of  "  developing "  and  "  fixing,"  becomes  a  permanent 
negative  picture  of  the  object.  When  a  "print"  is  made 
from  this  negative,  the  result  is  a  positive  picture. 

275.  The  Eye.  —  The  eye  is  like  a  small  photographic 
camera,  with  a  converging  lens,  a  dark  chamber,  and  a 
sensitive  screen.  Figure  248  is  a  vertical  section  through 
the  axis.  The  outer  covering,  or  sclerotic  coat  H,  is  a  thick 
opaque  substance,  except  in  front,  where  it  is  extended 

as  a  transparent 
coat,  called  the 
cornea  A.  Behind 
the  cornea  is  a  dia- 
phragm D,  consti- 
tuting the  colored 
part  of  the  eye,  or 
the  iris.  The  cir- 
cular opening  in 
the  iris  is  the  pupil, 


Fig.  248 


the  size  of  which 
changes  with  the 
intensity  of  light.  Supported  from  the  walls  of  the  eye, 
just  back  of  the  iris,  is  the  crystalline  lens  E,  a  transparent 
body  dividing  the  eye  into  two  chambers;  the  anterior 
chamber  between  the  cornea  and  the  crystalline  lens  is  a 
transparent  fluid  called  the  aqueous  humor,  while  the  large 
chamber  behind  the  lens  is  filled  with  a  jellylike  substance 
called  the  vitreous  humor.  The  choroid  coat  lines  the  walls 
of  this  posterior  chamber,  and  on  it  is  spread  the  retina,  a 
membrane  traversed  by  a  network  of  nerves,  branching 
from  the  optic  nerve  M.  The  choroid  coat  is  filled  with  a 
black  pigment,  which  serves  to  darken  the  cavity  of  the 
eye,  and  to  absorb  the  light  reflected  internally. 


OPTICAL  INSTRUMENTS  225 

276.  Sight.  —  When  rays  of  light  diverge  from  the  ob- 
ject and  enter  the  pupil  of  the  eye  they  form  an  inverted 
image  on  the  retina  (Fig.  249)  precisely  as  in  the  photo- 


Fig.  249 

graphic  camera.  In  place  of  the  sensitized  plate  is  the 
sensitive  retina,  from  which  the  stimulus  is  carried  to  the 
brain  along  the  optic  nerve. 

In  the  camera  the  distance  Between  the  lens  and  the 
screen  or  plate  must  be  adjusted  for  objects  at  different 
distances.  In  the  eye  the  corresponding  distance  is  fixed, 
and  the  adjustment  for  distinct  vision  is  made  by  uncon- 
sciously changing  the  curvature  of  the  front  surface  of 
the  crystalline  lens  by  means  of  the  ciliary  muscle  F,  Gr 
(Fig.  248).  This  capability  of  the  lens  of  the  eye  to 
change  its  focal  length  for  objects  at  different  distances 
is  called  accommodation. 

277.  The  Blind  Spot.  —  There  is  a  small  depression  where 
the  optic  nerve  enters  the  eye.  The  rest  of  the  retina  is 
covered  with  microscopic  rods  and  cones,  but  there  are 
none  in  this  depression,  and  it  is  insensible  to  light.  It 
is  accordingly  called  the  blind  spot.  Its  existence  can  be 
readily  proved  by  the  help  of  Fig.  250.  Hold  the  book 


Fig.  250 

with  the  circle  opposite  the  right  eye.     Now  close  the 
left  eye  and  turn  the  right  to  look  at  the  cross.     Move 


226  LIGHT 

the  book  toward  the  eye  from  a  distance  of  about  a  foot, 
and  a  position  will  readily  be  found  where  the  black  circle 
will  disappear.  Its  image  then  falls  on  the  blind  spot. 
It  may  be  brought  into  view  again  by  moving  the  book 
either  nearer  the  eye  or  farther  away. 

VII.    DISPERSION 

278.   Analysis  of   White  Light.      The    Solar  Spectrum. — 

Darken  the  room,  and  by  means  of  a  mirror  hinged  outside  the 
window,  reflect  a  pencil  of  sunlight  into  the  room.  Close  the  opening 

in  the  window  with  a 
piece  of  tin,  in  which 
is  cut  a  very  narrow 
vertical  slit.  Let  the 
ribbon  of  sunlight 
issuing  from  the  slit 
be  incident  obliquely 
on  a  glass  prism  (Fig. 
251).  A  many-col- 
ored band,  gradually 
changing  from  red 
at  one  end  through 
orange,  yellow,  green, 
blue,  to  violet  at  the 
P.  ~c- .  other,  appears  on  the 

screen.  If  a  converg- 
ing lens  of  about  30  cm.  focal  length  be  used  to  focus  an  image  of  the 
slit  on  the  screen,  and  the  prism  be  placed  near  the  principal  focus, 
the  colored  images  of  the  slit  will  be  more  distinct. 

This  experiment  shows  that  white  or  colorless  light  is 
a  mixture  of  an  infinite  numjber  of  differently  colored 
rays,  of  which  the  red  is  refracted  least  and  the  violet 
most.  The  brilliant  band  of  light  consists  of  an  indefinite 
number  of  colored  images  of  the  slit ;  it  is  called  the  solar 
spectrum,  and  the  opening  oufTor  separating  of  the  beam 
of  white  light  is  known  as  dispersion. 


DISPERSION 


227 


279.  Synthesis  of  Light.  —  Project  a  spectrum  of  sunlight  on  the 
screen.  Now  place  a  second  prism  like  the  first  behind  it,  but  re- 
versed in  position  (Fig.  252).  There 
will  be  formed  a  colorless  image,  slightly 
displaced  on  the  screen. 


The  second  prism  reunites  the 
colored  rays,  making  the  effect 

that  of  a  thick  plate  of  glass  (§  255).  The  recomposition 
"of  the  colored  rays  into  white  light  may  also  be  effected  by 
receiving  them  on  a  concave  mirror  or  a  large  convex  lens. 

280.  Chromatic  Aberration.  —  Let  a  beam  of  sunlight  into  the 
darkened  room  through  a  round  hole  in  a  piece  of  cardboard.     Project 
an  image  of  this  aperture  on  the  screen,  using  a  double-convex  lens  for 
the  purpose.  The  round  image  will  be  bordered  with  the  spectral  colors. 

This  experiment  shows  that  the  lens  refracts  the  rays 
of  different  colors  to  different  foci.     This  defect  in  lenses 

is  known  as  chromatic  aberration. 

A 

The  violet  rays,  being  more  re- 
frangible than  the  red,  will  have 
their  focus  nearer  to  the  lens  than 
the  red,  as  shown  in  Fig.  253,  where 
v  is  the  principal  focus  for  violet 

light  and  r  for  red.  If  a  screen  were  placed  at  a?,  the 
image  would  be  bordered  with  red,  and  if  at  y  with  violet. 

281.  The  Achromatic  Lens.  — With   a  prism  of  crown  glass 
project  a  spectrum  of  sunlight  on  the  screen,  and  note  the  length 
of  the  spectrum  when  the  prism  is  turned  to  give  the  least  deviation 
(§256).     Repeat  the    ex- 
periment with  a  prism  of 

flint  glass  having  the  same 
refracting  angle.  The  spec- 
trum formed  by  the  flint 
glass  will  be  about  twice  as 
long  as  that  given  by  crown  Fig.  254 


228 


LIGHT 


glass,  while  the  position  of  the  middle  of  the  spectrum  on  the  screen 
is  about  the  same  in  the  two  cases.  Now  use  a  flint  glass  prism  whose 
refracting  angle  is  half  that  of  the  crown  glass  one.  The  spectrum  is 
nearly  equal  in  length  to  that  given  by  the  crown  glass  prism,  but  the 
deviation  of  the  middle  of  it  is  considerably  less.  Finally,  place 
this  flint  glass  prism  in  a  reversed  position  against  the  crown  glass 
one  (Fig.  254).  The  image  of  the  aperture  is  no  longer  colored,  and 
the  deviation  is  about  half  that  produced  by  the  crown  glass  alone. 

In  1757  Dollond,  an  English  optician,  combined  a 
double-convex  lens  of  crown  glass  with  a  plano- 
concave lens  of  flint  glass  so  that  the  dispersion 
by  the  one  neutralized  that  due  to  the  other, 
while  the  refraction  was  reduced  about  half 
(Fig.  255).  Such  a  lens  or  system  of  lenses  is 

Fig.  255     called  achromatic,  since  images  formed  by  it  are 

not  fringed  with  the  spectral  colors. 

282.  The  Rainbow. —  Cement  a  crystallizing  beaker  12  or  15  cm. 
in  diameter  to  a  slate  slab.  Fill  the  beaker  with  water  through  a 
hole  drilled  in  the  slate.  Support  the  slate  in  a  vertical  plane  and 
direct  a  ribbon  of  white 
light  upon  the  beaker  at 
a  point  about  60°  above 
its  horizontal  axis,  as  SA 
(Fig.  256).  The  light 
may  be  traced  through 
the  water,  part  of  it 
issuing  at  the  back  at 
B  as  a  diverging  pencil, 
and  a  part  reflected  to  C 

and  issuing  as  spectrum  ^£-  ^° 

colors  along  CD.  If  other  points  of  incidence  be  tried,  the  colors 
given  by  the  reflected  portion  are  very  indistinct  except  at  70°  below 
the  axis.  After  refraction  at  this  point,  the  light  is  internally  reflected 
twice  and  then  emerges  in  front  as  a  diverging  pencil  of  spectrum  colors. 

The  experiment  shows  that  the  light  must  be  incident 
at  definite  angles  to  give  color  effects.     The  red  constitu- 


7 


DISPERSION 


229 


ent  of  white  light  incident  at  about  60°  keeps  together 
after  reflection  and  subsequent  refraction;  that  is,  the  red 
rays  are  practically  parallel  and  thus  have  sufficient  in- 
tensity to  produce  a  red  image.  The  same  is  true  of 
the  violet  light  incident  at  about  59°  from  the  axis.  The 
other  spectral  colors  arrange  themselves  in  order  between 
the  red  and  violet. 

When  sunlight  falls  in  this  manner  on  raindrops,  they 
disperse  the  light  and  the  spectral  colors  produced  form 
the  rainbow.  Two  bows 
are  often  visible,  the 
primary  and  the  second- 
ary. The  primary  bow 
is  the  inner  and  brighter 
one  formed  by  a  single 
internal  reflection ;  it  is 
distinguished  by  being 
red  on  the  outside  and 
violet  on  the  inside.  The 
secondary  low,  formed 
by  two  internal  reflec- 
tions, is  fainter,  and  has 
the  order  of  colors  re- 
versed. Figure  257  shows  the  relative  position  of  the  sun, 
the  observer,  and  the  raindrops  which  form  the  bows. 

283.  Continuous  Spectra.  —  Throw  on  a  screen  the  spectrum  of 
the  electric  arc,  using  preferably  for  the  purpose  a  hollow  prism  filled 
with  carbon  bisulphide.  The  spectrum  will  be  composed  of  colors 
from  red  at  one  end  through  orange,  yellow,  green,  blue,  and  violet 
at  the  other  without  interruptions  or  gaps. 

The  experiment  illustrates  continuous  spectra,  that  is, 
spectra  without  breaks  or  gaps  in  the  color  baud.  Solids, 


Fig.  257 


230 


LIGHT 


liquids,  and  dense  vapors  and  gases,  when  heated  to  incan- 
descence, give  continuous  spectra. 

284.  Discontinuous  Spectra.  —  Project  on  the  screen  the  spec- 
trum of  the  electric  light.     Place  in  the  arc  a  few  crystals  of  sodium 
nitrate.     The  intense  heat  will  vaporize  the  sodium,  and  a  spectrum 
will  be  obtained  consisting  of  bright  colored  lines,  one  red,  one  yellow, 
three  green,  and  one  violet,  the  yellow  being  most  prominent. 

The  experiment  illustrates  discontinuous  or  bright  line 
spectra,  that  is,  spectra  consisting  of  one  or  more  bright 
lines  of  coJLer  separated  by  dark  spaces.  Rarefied  gases 
and  vapors,  when  heated  to  incandescence,  give  discontinuous 
spectra. 

285.  Absorption  Spectra.  —  Project  on  the  screen  the  spectrum 
of  the  electric  light.     Between  the  lamp  and  the  slit  S  (Fig.  258) 
vaporize  metallic  sodium  in  an  iron  spoon  so  placed  that  the  white 

light  passes  through 
the  heated  sodium 
vapor  before  disper- 
sion by  the  prism. 
A  dark  line  will  ap- 
pear on  the  screen 
in  the  yellow  of 
the  spectrum  at  the 
place  where  the 
bright  line  was  ob- 
tained in  the  pre- 
ceding experiment. 


ment  illustrates 

an  absorption,  reversed,  or  dark  line  spectrum.  The  dark 
line  is  produced  by  the  absorption  of  the  yellow  light 
by  sodium  vapor.  Gases  and  vapors  absorb  light  of  the 
same  refrangibility  as  they  emit  at  a  higher  temperature. 


DISPERSION  231 

286.  The  Fraunhofer  Lines.  —  Show  on  the  screen  a  carefully 
focused  spectrum  of  sunlight.  Several  of  the  colors  will  appear  crossed 
with  fine  dark  lines  (Fig.  259). 


iBC 


Eb 


Bed    Orange  Fellow  Green      Blue        Indigo  Twlet 

Fig.  259 

Fraunhofer  was  the  first  to  notice  that  some  of  these 
lines  coincide  in  position  with  the  bright  lines  of  certain 
artificial  lights.  He  mapped  no  less  than  576  of  them, 
and  designated  the  more  important  ones  by  the  letters 
A,  B,  0,  D,  E,  F,  G-,  H,  the  first  in  the  extreme  red  and 
the  last  in  the  violet.  For  this  reason  they  are  referred 
to  as  the  Fraunhofer  lines.  In  recent  years  the  number 
of  these  lines  has  been  found  to  be  practically  unlimited. 

In  the  last  experiment  it  was  shown  that  sodium  vapor 
absorbs  that  part  of  the  light  of  the  electric  arc  which  is 
of  the  same  refrangibility  as  the  light  emitted  by  the  vapor 
itself.  Similar  experiments  with  other  substances  show 
that  every  substance  has  its  own  absorption  spectrum. 
These  facts  suggested  the  following  explanation  of  the 
Fraunhofer  lines :  The  heated  nucleus  of  the  sun  gives 
off  light  of  all  degrees  of  refrangibility.  Its  spectrum 
would  therefore  be  continuous,  were  it  not  surrounded 
by  an  atmosphere  of  metallic  vapors  and  of  gases,  which 
absorb  or  weaken  those  rays  of  which  the  spectra  of  these 
vapors  consist.  Hence,  the  parts  of  the  spectrum  which 
would  have  been  illuminated  by  those  particular  rays  have 
their  brightness  diminished,  since  the  rays  from  the  nucleus 
are  absorbed,  and  the  illumination  is  due  to  the  less  intense 
light  coming  from  the  vapors.  These  absorption  lines  are 


232 


LIGHT 


not  lines  of  no  light,  but  are  lines  of  diminished  bright- 
ness, appearing  dark  by  contrast  with  the  other  parts  of 
the  spectrum. 

287.  The  Spectroscope. — The  commonest  instrument  for 
viewing  spectra  is  the  spectroscope  (Fig.  260).  In  one 
of  its  simplest  forms  it  consists  of  a  prism  A,  a  telescope  B, 
and  a  tube  called  the  collimator  C,  carrying  an  adjustable 


Fig.  260. 

slit  at  the  outer  end  _D,  and  a  converging  lens  at  the  other  E, 
to  render  parallel  the  diverging  rays  coming  from  the  slit. 
The  slit  must  therefore  be  placed  at  the  principal  focus  of 
the  converging  lens.  To  mark  the  deviation  of  the  spec- 
tral lines,  there  is  provided  on  the  supporting  table  a 
divided  circle  F,  which  is  read  by  the  aid  of  verniers  and 
reading  microscopes  attached  to  the  telescope  arm. 

The  applications  of  the  spectroscope  are  many  and  various.     By  an 
examination  of  their  absorption  spectra,  normal  and  diseased  blood 


COLOR 


233 


are  easily  distinguished,  the  adulteration  of  substances  is  detected, 
and  the  chemistry  of  the  stars  is  approximately  determined.  Figure 
261  shows  the  agreement  of  a  number  of  the  spectral  lines  of  iron 
with  Fraunhofer  lines  in  the  solar  spectrum ;  they  indicate  the  pre- 
sence of  iron  vapor  in  the  atmosphere  of  the  sun. 


HIIIHIIII 


Fig.  261 


VIII.     COLOR 

288.  The  Wave  Length  of  light  determines  its  color. 
Extreme  red  is  produced  by  the  longest  waves,  and  ex- 
treme violet  by  the  shortest.  The  following  are  the  wave 
lengths  for  the  principal  Fraunhofer  lines  in  air  at  20°  C. 
and  760  mm.  pressure  :  — 


5269  mm. 
4861  mm. 
4293  mm. 
3968  mm. 


In  white  light  the  number  of  colors  is  infinite,  and  they 
pass  into  one  another  by  imperceptible  gradations  of  shade 
and  wave  length.  Color  stands  related  to  light  in  the  same 
way  that  pitch  does  to  sound.  In  most  artificial  lights  cer- 
tain colors  are  either  feeble  or  wanting.  Hence,  artificial 
lights  are  not  generally  white,  but  each  one  is  character- 
ized by  the  color  that  predominates  in  its  spectrum. 


A 
B 
C 

Do 

Dark  Red   .     0.0007621  mm. 
Red    .     .     .             6884  mm. 
Orange  .     .              6563  mm. 
Yellow   .     .             5896  mm. 
5890  mm. 

Ei  Light  Green 

F  Blue  .  .  . 
G  Indigo  .  . 
H,  Violet 

234  LIGHT 

289.  Color  of  Opaque  Bodies.  —  Project  the  solar  spectrum  on  a 
white  screen.     Hold  pieces  of  colored  paper  or  cloth  successively  in 
different  parts  of  the  spectrum.     A  strip  of  red  flannel  appears  bril- 
liantly red  in  the  red  part  of  the  spectrum,  and  black  elsewhere ;  a 
blue  ribbon  is  blue  only  in  the  blue  part  of  the  spectrum,  and  a  piece 
of  black  paper  is  black  in  every  part  of  the  spectrum. 

The  experiment  shows  that  the  color  of  a  body  is  due  both 
to  the  light  that  it  receives  and  the  light  that  it  reflects ; 
that  a  body  is  red  because  it  reflects  chiefly,  if  not  wholly, 
the  red  rays  of  the  light  incident  upon  it,  the  others  being 
absorbed  wholly  or  partly  at  its  surface.  It  cannot  be  red 
if  there  is  no  red  light  incident  upon  it.  In  the  same  way 
a  body  is  white  if  it  reflects  all  the  rays  in  about  equal 
proportions,  provided  white  light  is  incident  upon  it.  So 
it  appears  that  bodies  have  no  color  of  their  own,  since 
they  exhibit  no  color  not  already  present  in  the  light 
which  illuminates  them.  This  truth  is  illustrated  by  the 
difficulty  experienced  in  matching  colors  by  artificial  lights, 
and  by  the  changes  in  shade  some  fabrics  undergo  when 
taken  from  sunlight  into  gaslight.  Most  artificial  lights 
are  deficient  in  blue  and  violet  rays;  and  hence  all  com- 
plex colors,  into  which  blue  or  violet  enters,  as  purple  and 
pink,  change  their  shade  when  viewed  by  artificial  light. 

290.  Color  of  Transparent  Bodies.  —  Throw  the  spectrum  of 
the  sun  or  of  the  arc  light  on  the  screen.     Hold  across  the  slit  a  flat 
bottle  or  cell  filled  with  a  solution  of  ammoniated  oxide  of  copper.1 
The  spectrum  below  the  green  will  be  cut  off.     Substitute  a  solution 
of  picric  acid,  and  the  spectrum  above  the  green  will  be  cut  off.    Place 
both  solutions  across  the  slit  and  the  green  alone  remains.      It  is  the 
only  color  transmitted  by  both  solutions.     In  like  manner,  blue  glass 
cuts  off  the  less  refrangible  part  of  the  spectrum,  ruby  glass  cuts  off 
the  more  refrangible,  and  the  two  together  cut  off  the  whole. 

1  It  is  prepared  by  adding  ammonia  to  a  solution  of  copper  sulphate, 
until  the  precipitate  at  first  formed  is  dissolved. 


COLOR 


235 


This  experiment  shows  that  the  color  of  a  transparent 
body  is  determined  by  the  colors  that  it  absorbs.  It  is 
colorless  like  glass  if  it  absorbs  all 
colors  in  like  proportion,  or  absorbs 
none;  but  if  it  absorbs  some  colors 
more  than  others,  its  color  is  due 
to  the  mixed  impression  produced 
by  the  various  colors  passing 
through  it. 


291.   Mixing  Colored  Lights.  —  Out 

of  colored  papers  cut  several  disks,  about 
15  cm.  in  diameter,  with  a  hole  at  the 
center  for  mounting  them  on  the  spindle 
of  a  whirling  machine  (Fig.  262),  or  for 
slipping  them  over  the  handle  of  a  heavy 
spinning  top.  Slit  them  along  a  radius 
from  the  circumference  to  the  center,  so 
that  two  or  more  of  them  can  be  placed  Fig.  262 

together,  exposing  any  proportional  part 

of  each  one  as  desired  (Fig.  263).  Select  seven  disks,  whose  colors 
most  nearly  represent  those  of  the  solar  spectrum ;  put  them  to- 
gether so  that  equal  portions  of  the  colors  are  exposed.  Clamp 
on  the  spindle  of  the  whirling  machine  and  rotate  them  rapidly. 
"When  viewed  in  a  strong  light  the  color  is  an  impure  white  or 
gray. 

This  method  of  mixing  colors  is  based  on  the  physio- 
logical fact  that  a  sensation  lasts  longer  than  the  stim- 
ulus   producing    it.       Before    the 
sensation  caused  by  one   stimulus 
has  ceased,  the  disk  has  moved,  so 
that  a  different  impression  is  pro- 
Fig.  263  duced.      The   effect   is   equivalent 
to  superposing  the  several  colors  on  one  another  at  the 
same  time. 


236  LIGHT 

292.  Three  Primary  Colors.  —  If  red,  green,  and  blue,  or 
violet  disks  are  used,  as  in  §  291,  exposing  equal  portions, 

gray  or  impure  white  is  obtained 
when  they  are  rapidly  rotated. 
If  any  two  colors  standing  op- 
posite each  other  in  Fig.  264  are 
used,  the  result  is  white ;  and 
if  any  two  alternate  ones  are 
used,  the  result  is  the  interme- 
diate one.  By  using  the  red, 
the  green,  and  the  violet  disks, 
and  exposing  in  different  pro- 
F.  264  portions,  it  has  been  found 

possible  to  produce  any  color 
of  the  spectrum.  This  fact  suggested  to  Dr.  Young  the 
theory  that  there  are  only  three  primary  color  sensations, 
and  that  our  recognition  of  different  colors  is  due  to  the 
excitation  of  these  three  in  varying  degrees. 

The  color  top  is  a  standard  toy  provided  with  colored 
paper  disks,  like  those  of  Fig.  262.  When  red,  green, 
and  blue  disks  are  combined  so  as  to  show  sectors  of  equal 
size,  the  top,  when  spinning  in  a  strong  light,  appears  to 
be  gray.  Gray  is  a  white  of  low  intensity.  •  The  colors 
of  the  disks  are  those  of  pigments,  and  they  are  not  pure 
red,  green,  and  blue. 

293.  Three-color   Printing.  —  The    frontispiece    in    this 
book  illustrates  a  three-color  print  of  much  interest.     Such 
a  print  is  made  up  of  very  fine  lines  and  dots  of  the  three 
pigments,  red,  yellow,  and  blue  ;  the  various  colors  in  the 
picture  are  mixtures  of  these  three  with  the  white  of  the 
paper.     The  greens  come  chiefly  from  the  overlapping  and 
mixture  of  the  yellow  and  blue  pigments. 


COLOR  237 

The  process  is  briefly  as  follows  :  Three  negatives  of 
the  same  original  are  taken  through  transparent  screens  of 
red,  green,  and  blue,  and  each  is  crossed  by  fine  lines  or 
dots.  Copper  plates  are  made  from  the  negatives,  and  each 
plate  is  inked  for  printing  with  an  ink  of  a  color  which 
gives  white,  when  mixed  with  the  color  of  the  screen 
through  which  the  negative  was  taken.  Thus,  the  plate 
made  with  the  red  screen  is  printed  with  greenish  blue 
ink;  those  taken  with  the  green  and  blue  or  violet  screens 
are  printed  with  crimson,  red,  or  yellow  ink,  respectively. 
In  the  frontispiece  the  first  plate  was  printed  with  yellow, 
the  second  with  yellow  and  then  with  red,  and  the  third 
with  all  three. 

294.  Complementary  Colors.  —  Any  two  colors  whose  mix- 
ture produces  on  the  eye  the  impression  of  white  light  are 
called  complementary.  Thus,  red  and  bluish  green  are 
complementary;  also  orange  and  light  blue.  When  com- 
plementary colors  are  viewed  next  to  each  other,  the  effect 
is  a  mutual  heightening  of  color  impressions. 

Complementary  colors  may  be  seen  by  what  is  known  as  retinal 
fatigue.  Cut  some  design  out  of  paper,  and  paste  it  on  red  glass. 
Project  it  on  a  screen  in  a  dark  room.  Look  steadily  at  the  screen  for 
several  seconds,  and  then  turn  up  the  lights.  The  design  will  appear 
on  a  pale  green  ground. 

This  experiment  shows  that  the  portion  of  the  retina 
on  which  the  red  light  falls  becomes  tired  of  red,  and 
refuses  to  convey  as  vivid  a  sensation  of  red  as  of  the 
other  colors,  when  less  intense  white  light  is  thrown  on 
it.  But  it  retains  its  sensitiveness  in  full  for  the  rest  of 
white  light,  and  therefore  conveys  to  the  brain  the  impres- 
sion of  white  light  with  the  red  cut  out ;  that  is,  of  the 
complementary  color,  green. 


238  LIGHT 

295.  Mixing  Pigments.  —  Draw  a  broad  line  on  the  blackboard 
with  a  yellow  crayon.  Over  this  draw  a  similar  band  with  a  blue 
crayon.  The  result  will  be  a  band  distinctly  green. 

The  yellow  crayon  reflects  green  light  as  well  as  yellow, 
and  absorbs  all  the  other  colors.  The  blue  crayon  reflects 
green  light  along  with  the  blue,  absorbing  all  the  others. 
Hence,  in  superposing  the  two  chalk  marks,  the  mixture 
absorbs  all  but  the  green.  The  mark  on  the  board  is 
green,  because  that  is  the  only  color  that  survives  the 
double  absorption.  In  mixing  pigments,  the  resulting 
color  is  the  residue  of  a  process  of  successive  absorptions. 
If  the  spectral  colors,  blue  and  yellow,  are  mixed,  the 
product  is  white  instead  of  green.  So  we  see  that  a  mix- 
ture of  colored  lights  is  a  very  different  thing  from  a 
mixture  of  pigments. 


IX.     INTERFERENCE  AND  DIFFRACTION 

296.  Newton's  Rings.  —  Press  together  at  their  center  two  small 
pieces  of  heavy  plate  glass,  using  a  small  iron  clamp  for  the  purpose. 
Then  look  obliquely  at  the  glass ;  curved  bands  of  color  may  be  seen 
surrounding  the  point  of  greatest  pressure. 

This  experiment  is  like  one  performed  by  Newton  while 
attempting  to  determine  the  relation  between  the  colors 

in  the  soap  bubble  and  the  thick- 
ness of  the  film.     He  used  a  plano- 
convex  lens  of  long  focus  resting 

G  f  F  G 

on  a  plate  of  plane  glass.     Figure 
265  shows  a  section  of  the  appa- 
ratus.    Between  the  lens  and  the  plate  there  is  a  wedge- 
shaped  film  of  air,  very  thin,  and  quite  similar  to  that 
formed  between  the  glass  plates  in  the  above  experiment. 


INTERFERENCE  AND  DIFFRACTION 


239 


Fig.  266 


If  the  glasses  are  viewed  by  reflected  light,  there  is  a 
dark  spot  at  the  point  of  contact,  surrounded  by  several 
colored  rings  (Fig.  266) ;  but  if  viewed  by  transmitted 
light,  the  colors  are  complementary  to  those  seen  by  re- 
flection (§  294).  The  explanation  is  to  be  found  in  the 
interference  of  two  sets  of  waves, 
one  reflected  internally  from  the 
curved  surface  AGB,  and  the 
other  from  the  surface  DOE,  on 
which  it  presses.  If  light  of  one 
color  is  incident  on  AB,  a  portion 
will  be  reflected  from  ACB,  and 
another  portion  from  D  OE.  Since 
the  light  reflected  from  D  OE  has 
traveled  farther  by  twice  the 
thickness  of  the  air  film  than  that 
from  ACB,  and  the  film  gradually  increases  in  thickness 
from  O  outward,  it  follows  that  at  some  places  the  two 
reflected  portions  will  meet  in  like  phase,  and  at  others  in 
opposite  phase,  causing  a  strengthening  of  the  light  at  the 
former,  and  extinction  of  it  at  the  latter.  If  red  light  be 
used,  the  appearance  will  be  that  of  a  series  of  concentric 
circular  red  bands  separated  by  dark  ones,  each  shading 
off  into  the  other.  If  violet  light  be  employed,  the  colored 
bands  will  be  closer  together  on  account  of  the  shorter 
wave  length.  Other  colors  will  give  bands  intermediate 
in  diameter  between  the  red  and  violet.  From  this  it  fol- 
lows that  if  the  glasses  be  illuminated  by  white  light,  at 
every  point  some  one  color  will  be  destroyed.  The  other 
colors  will  be  either  weakened  or  strengthened,  depending 
on  the  thickness  of  the  air  film  at  the  point  under  consid- 
eration, the  color  at  each  point  being  the  result  of  mixing 
a  large  number  of  colors  in  unequal  proportions.  Hence, 


240  LIGHT 

the  point   C  will  be  surrounded  by  a  series  of  colored 
bands.1 

The  colors  of  the  soap  bubble,  of  oil  on  water,  of  heated 
metals  which  easily  oxidize,  of  a  thin  film  of  varnish,  and 
of  the  surface  of  very  old  glass,  are  all  caused  by  the  in- 
terference of  light  reflected  from  the  two  surfaces  of  a 
very  thin  film. 

297.  Diffraction.  —  Place  two  superposed  pieces  of  perforated 
cardboard  in  front  of  the  condenser  of  the  projection  lantern.  The 
projected  images  of  the  very  small  holes,  as  one  piece  is  moved  across 
the  other,  are  fringed  with  the  spectral  colors. 

With  a  fine  diamond  point  rule  a  number  of  equidistant  parallel 
lines  very  close  together  on  glass.  They  compose  a  transparent  dif- 
fraction grating.  Substitute  this  for  the  prism  in  projecting  the  spec- 
trum of  sunlight  or  of  the  arc  light  on  the  screen  (§  278).  There  will 
be  seen  on  the  screen  a  central  image  of  the  slit,  and  on  either  side  of 
it  a  series  of  spectra.  Cover  half  of  the  length  of  the  slit  with  red 
glass  and  the  other  half  with  blue.  There  will  now  be  a  series  of  red 
images  and  also  a  series  of  blue  ones,  the  red  ones  being  farther  apart 
than  the  blue.  Lines  ruled  close  together  on  smoked  glass  may  be 
used  instead  of  a  "  grating." 

These  experiments  illustrate  a  phenomenon  known  as 
diffraction.  The  colored  bands  are  caused  by  the  inter- 
ference of  the  waves  of  light  which  are  propagated  in  all 
directions  from  the  fine  openings.  The  effects  are  visible 
because  the  transparent  spaces  are  so  small  that  the  inten- 
sity of  the  direct  light  from  the  source  is  largely  reduced. 
Diffraction  gratings  are  also  made  to  operate  by  reflecting 
light.  Striated  surfaces,  like  mother-of-pearl,  changeable 

1  The  light  from  ACB  differs  in  phase  half  a  wave  length  from  that 
reflected  from  DE,  because  the  former  is  reflected  in  an  optically  dense 
medium  next  to  a  rare  one,  and  the  latter  in  an  optically  rare  medium 
next  to  a  dense  one.  This  phase  difference  is  additional  to  the  one  above 
described. 


QUESTIONS  241 

silk,  and  the  plumage  of  many  birds,  owe  their  beautiful 
changing  colors  to  interference  of  light  by  diffraction. 

Questions 

1.  Why  is  the  flint  glass  of  an  achromatic  lens  the  diverging  part 
of  the  combination  instead  of  the  converging? 

2.  Why  is  the  rainbow  circular? 

3.  Do  different  persons  see  the  same  rainbow  ? 

4.  Why  is  the  rainbow  not  seen  at  midday? 

5.  In  projecting  pictures  on  a  screen,  why  should  the  screen  be 
white  ? 

6.  Of  which  case  of  images  by  lenses  is  the  projecting  lantern  an 
application  ? 

7.  In  enlarging  a  negative  by  photography  where  must  the  nega- 
tive be  placed  with  respect  to  the  lens? 

8.  Why  do  flowers  that  are  purple  by  sunlight  look  red  by  lamp- 
light? 

9.  Account  for  the  color  on  a  plate  of  glass  when  it  is  brushed 
over  with  alcohol. 

10.   Account  for  the  crossed  bands  of  colors  seen  by  looking  through 
a  silk  umbrella  at  an  arc  electric  light. 


CHAPTER  IX 

HEAT 
I.   HEAT  AND  TEMPERATURE 

298.  Nature  of  Heat.  —  For  a  long  time  it  was  believed 
that  heat  was  a  subtle  and  weightless  fluid  that  entered 
bodies  and  possibly  combined  with  them.  This  fluid  was 
called  caloric.  About  the  beginning  of  the  last  century 
certain  experiments  of  Count  Rumford  and  Sir  Humphry 
Davy  demonstrated  that  the  caloric  theory  of  heat  was 
no  longer  tenable ;  and  finally  about  the  middle  of  the 
century,  when  Joule  proved  that  a  definite  amount  of 
mechanical  work  is  equivalent  to  a  definite  amount  of 
heat,  it  became  evident  that  heat  is  a  form  of  molecular 
energy. 

The  modern  kinetic  theory,  briefly  stated,  is  as  follows  : 
The  molecules  of  a  body  have  a  certain  amount  of  inde- 
pendent motion,  generally  very  irregular.  Any  increase 
in  the  energy  of  this  motion  shows  itself  in  additional 
warmth,  and  any  decrease  by  the  cooling  of  the  body. 
The  heating  or  the  cooling  of  a  body,  by  whatever 
process,  is  but  the  transference  or  the  transformation  of 
energy. 


Temperature.  —  If  we  place  a  mass  of  hot  iron  in 
contact  with  a  cold  one,  the  latter  becomes  warmer  and  the 
former  cooler,  the  heat  flowing  from  the  hot  body  to  the 

242 


HEAT  AND   TEMPERATURE  243 

cold  one.  The  two  bodies  are  said  to  differ  in  temperature 
or  "  heat  level,"  and  when  they  are  brought  in  contact 
there  is  a  flow  of  heat  from  the  one  of  higher  temperature 
to  the  one  of  lower  till  thermal  (heat)  equilibrium  is 
established.  Temperature  is  the  thermal  condition  of  a 
body  which  determines  the  transfer  of  heat  between  it 
and  any  body  in  contact  with  it.  This  transfer  is  always 
from  the  body  of  higher  temperature  to  the  one  of  lower. 
Temperature  is  a  measure  of  the  degree  of  hotness;  it 
depends  solely  on  the  kinetic  energy  of  the  molecules  of 
the  body.  Temperature  must  be  distinguished  from  quan- 
tity of  heat.  The  water  in  a  pint  cup  may  be  at  a  much 
higher  temperature  than  the  water  in  a  lake,  yet  the  latter 
contains  a  vastly  greater  quantity  of  heat,  owing  to  the 
greater  quantity  of  water. 

300.  Measuring  Temperature.  —  Fill  three  basins  with  moder- 
ately hot  water,  cold  water,  and  tepid  water  respectively.  Hold  one 
hand  in  the  first,  and  the  other  in  the  second  for  a  short  time ;  then 
transfer  both  quickly  to  the  tepid  water.  It  will  feel  cold  to  the 
hand  that  has  been  in  hot  water  and  warm  to  the  other.  Hold  the 
hand  successively  against  a  number  of  the  various  objects  in  the 
room,  at  about  the  same  height  from  the  floor.  Metal,  slate,  or  stone 
objects  will  feel  colder  than  those  of  wood,  even  when  side  by  side 
and  of  the  same  temperature. 

These  experiments  show  that  the  sense  of  touch  does 
not  give  accurate  information  regarding  the  relative 
temperature  of  bodies,  and  some  other  method  must  be 
resorted  to  for  reliable  measurement.  The  one  most  ex- 
tensively used  is  based  on  the  regular  increase  in  the 
volume  of  a  body  attending  a  rise  in  its  temperature. 
This  method  is  illustrated  by  the  common  mercurial  ther- 
mometer. 


244 


HEAT 


II.    THE  THERMOMETER 

301.  The  Thermometer.  —  The  common  mercurial  ther- 
mometer consists  of  a  capillary  glass  tube  of  uniform  bore, 
on  one  end  of  which  is  blown  a  bulb,  either  spherical  or 
cylindrical  (Fig.  267).  Part  of  the  air  is  expelled  by 
heating,  and  while  in  this  condition  the  open 
end  of  the  tube  is  dipped  into  a  vessel  of 
pure  mercury.  As  the  tube  cools,  mercury 
is  forced  into  the  tube  by  atmospheric  pres- 
sure. Enough  mercury  is  introduced  to  fill 
the  bulb  and  part  of  the  tube  at  the  lowest 
temperature  which  the  thermometer  is  de- 
signed to  measure.  Heat  is  now  applied  to 
the  bulb  till  the  expanded  mercury  fills  the 
tube ;  the  end  is  then  closed  in  the  blowpipe 
flame.  The  mercury  contracts  as  it  cools, 
leaving  a  vacuum  at  the  top  of  the  tube. 

302.    Necessity  of  Fixed  Points.  —  No   two 

thermometers  are  likely  to  have  bulbs  and 
stems  of  the  same  capacity.  Consequently, 
the  same  increase  of  temperature  will  not 
produce  equal  changes  in  the  height  of  the  mercury.  If, 
then,  the  same  scale  were  attached  to  all  thermometers, 
their  indications  would  differ  so  widely  that  the  results 
would  be  worthless.  Hence,  if  thermometers  are  to  be 
compared,  corresponding  divisions  on  the  scale  of  different 
instruments  must  indicate  the  same  temperature.  This  may 
be  done  by  graduating  every  thermometer  by  comparison 
with  a  standard,  an  expensive  proceeding  and  for  many 
purposes  unnecessary,  since  mercury  has  a  nearly  uniform 
rate  of  expansion.  If  two  points  are  marked  on  the  stem, 
the  others  can  be  obtained  by  dividing  the  space  between 


Fig.  267 


THE  THERMOMETER 


245 


them  into  the  proper  number  of  equal  parts.  Investigations 
have  made  it  certain  that  under  a  constant  pressure  the 
temperature  of  melting  ice  and  that  of  steam  are  invariable. 
Hence,  the  temperature  of  melting  ice  and  that  of  steam 
under  a  pressure  of  76  cm.  of  mercury  (one  atmosphere) 
have  been  chosen  as  the  fixed  points  on  a  thermometer. 

303.  Marking  the  Fixed  Points.  —  The   thermometer   is 
packed  in  finely  broken  ice,  as  far  up  the  stem  as  the  mer- 
cury extends.     The  containing  vessel  (Fig.  268)  has  an 
opening  at  the  bottom  to  let  the 

water  run  out.  After  standing 
in  the  ice  for  several  minutes 
the  top  of  the  thread  of  mer- 
cury is  marked  on  the  stem. 
This  is  called  the  freezing  point. 
The  boiling  point  is  marked 
by  observing  the  top  of  the  mer- 
curial column  when  the  bulb 
and  stem  are  enveloped  in  steam 
(Fig.  269)  under  an  atmospheric 
pressure  of  76  cm.  (29.92  in.)- 
If  the  pressure  at  the  time 
is  not  76  cm.,  then  a  cor- 
rection must  be  applied,  the  Fi£-  268  Fis- 269 
amount  being  determined  by  the  approximate  rule  that 
the  temperature  of  steam  rises  0.1°  C.  for  every  increase 
of  2.71  mm.  in  the  barometric  reading,  near  100°  C. 

304.  Thermometer  Scales.  —  The   distance   between   the 
fixed   points  is   divided  into   equal   parts  called  degrees. 
The  number  of  such  parts  is  wholly  arbitrary,  and  several 
different   scales  have   been  introduced.     Three  of   these 
are  in  use  at  the  present  time :  the  Fahrenheit,  the  Genii- 


246  HEAT 

grade,  and  the  Reaumur.  The  Fahrenheit  scale  was 
introduced  by  Fahrenheit  about  1714,  and  is  the  one 
in  common  use  in  all  English-speaking  countries.  For 
some  unknown  reason  he  marked  the  freezing  point  at 
32°  above  the  zero  of  the  scale,  and  the  boiling  point  at 
212°,  dividing  the  space  between  into  180  equal  parts. 

The  Centigrade  scale  was  designed  by  Celsius  about 
1742.  It  differs  from  the  Fahrenheit  in  making  the 
freezing  point  0°  and  the  boiling  point  100°,  the  space 
between  being  divided  into  100  equal  parts.  This  is  the 
one  in  general  use  among  scientific  men. 

The  Reaumur  scale  marks  the  freezing  point  0°  and  the 
boiling  point  80°.  This  is  the  household  scale  on  the 
continent  of  Europe ;  in  this  country  its  use  is  restricted 
to  breweries.  Each  of  these  scales  is  extended  beyond 
the  fixed  points  as  far  as  desired.  The  divisions  below 
0°  are  read  as  negative ;  for  example,  —  10°  signifies  10 
degrees  below  zero.  The  reading  according  to  any  par- 
ticular scale  is  indicated  by  affixing  the  initial  letter  of 
the  name;  for  example,  5°  F.,  5°  C.,  and  5°  R.  signify 
5  degrees  above  zero  on  the  Fahrenheit,  Centigrade,  and 
Reaumur  scales  respectively. 

305.  The  Three  Scales  Compared.  —  In  Fig.  270  AB  is  a 
thermometer  with  three  scales  attached,  P  is  the  head 

of   the  mercury  col-    A  p         \      n 

umn,  and  F,  (7,  and 
R   are   the   readings 


Fahrenheit 
Centigrade 


on  the  scales  respec-    Reaun^ 


112 


100 


tively.     On  the  Fah- 

I/M.          i        A  -D  Fig.  270. 

renheit   scale   AB  — 

180  and  AP  =  F '—  32,  since  the  zero  is  32  spaces  below 
A;    on  the  Centigrade  JJ?=100   and  AP=  0;    on  the 


THE  THERMOMETER 


247 


Reaumur  AB 
to  AB  is 


80  and  AP  =  R.     Then  the  ratio  of  AP 


_  39        n     '72 

—  -  -  =  ——  =  —  .     By  substituting  the  read- 
180         1UU      oU 

ing  on  any  one  scale  in  this  equation  the  equivalent  on 
either  of  the  other  scales  is  easily  obtained.  For  ex- 
ample, if  it  is  required  to  express  68°  F.  on  the  Centi- 

grade scale,  then  68~    2 


180 


100 


whence  (7=20°. 


>/  )c 
*' 


\  6-oTC- 

306.  Limitations  of  the  Mercurial  Thermometer.  —  As  mer- 
cury  freezes   at    —  38.8°    C.,    it  cannot  be   used   as   the 
thermometric  substance  below  this  temperature. 

For  temperatures  below  —  38°  C.  alcohol  is  sub- 
stituted for  mercury.  Under  a  pressure  of  one 
atmosphere  mercury  boils  at  about  350°  C.  For 
temperatures  approaching  this  value  and  up  to 
about  550°  C.  the  thermometer  stem  is  filled  with 
pure  nitrogen  under  pressure.  The  pressure  of 
the  gas  keeps  the  mercury  from  boiling  (§  331). 

307.  The    Clinical    Thermometer.  —  The    clinical 
thermometer  is  a  sensitive   instrument   of   short 
range  for  indicating  the  temperature  of  the  human 
body.     It  is  usually  graduated  from  95°  to  110°  F., 
or   from    35°  to  45°  C.     There  is  a    constriction 
in   the   tube    just   above    the    bulb  (Fig.    271), 
which  causes  the  thread  of  mercury  to  break  at 
that  point  when  the  temperature  begins  to  fall, 
leaving  the  top  of  the  separated  thread  to  mark 
the   highest  temperature   registered.     A   sudden     lg* 
jerk  or  tapping  of  the  thermometer  forces  the  mercury 
down  past  the  constriction  and  sets  it  for  a  new  reading. 


248  HEAT 

Questions  and  Problems 

1.  Why  should  the  tube  of  a  thermometer  be  of  uniform  bore  ? 

2.  Is  it  correct  to  speak  of  one  temperature  as  twice  that  of  an- 
other? 

3.  Why  may  thermometers  differ  in  length  and  still  measure  be- 
tween the  same  extremes  of  temperature? 

4.  Why  do  thermometers  have  a  bulb  on   one  end?     Would  a 
closed  uniform  tube  answer  just  as  well? 

5.  What  would  be  the  effect  on  the  indications  of  a  thermometer 
if  the  glass  expanded  the  same  as  the  mercury? 

6.  Convert  into   equivalent   readings   on  the  Centigrade  scale: 
98°  F.,  -  40°  F.,  68°  F. 

7.  Convert  into  equivalent  readings   on    the    Fahrenheit   scale: 
36°  C.,  -  40°  C.,  29°  C. 

8.  The  lowest  temperature  yet  obtained  is  claimed  to  be  —  271.3° 
C.     What  would  this  be  on  the  Fahrenheit  scale  ? 

9.  The   melting  points   of  iron   and   copper  are  2737°   F.   and 
1943°  F.  respectively.      Express   these  temperatures   in    Centigrade 
degrees. 

10.  A  correct  Fahrenheit  thermometer  registers  the  temperature  of 
a  room  as  70°;  a  faulty  Centigrade  thermometer  reads  20°.     Find  the 
error  of  the  latter. 

11.  When  the  barometric  pressure  is  74  cm.,  what  is  the  boiling 
point? 

12.  A  certain  Centigrade  thermometer  registers  2°  in  melting  ice 
and  100°  in  steam  under  normal  atmospheric  pressure.     What  is  the 
correct  value  of  a  temperature  of  25°  as  given  by  this  instrument? 

III.    EXPANSION 

308.  Expansion  of  Solids.  —  Insert  a  long  knitting  needle  A 
in  a  block  of  wood  so  as  to  stand  vertically  (Fig.  272).  A  second 
needle  D  is  supported  parallel  to  the  first  by  means  of  a  piece  of  cork 
or  wood  C.  The  lower  end  of  D  just  touches  the  mercury  in  the  cup 
H.  An  electric  circuit  is  made  through  the  mercury,  the  needle, 


EXPANSION 


249 


an  electric  battery,  and  the  bell  B,  as  shown.  Now  apply  a  Bunsen 
flame  to  A ;  D  will  be  lifted  out  of  the  mercury  and  the  bell  will 
stop  ringing.  Then  heat  D  or  cool  A,  and  the  contact  of  D 
with  the  mercury  will  be 
renewed  as  shown  by  the 
ringing  of  the  bell. 

This  experiment 
shows  that  solids  ex- 
pand in  length  when 
heated  and  contract 
when  cooled.  To  this 
rule  of  expansion  there 
are  a  few  exceptions, 
notably  iodide  of  silver 
and  stretched  india- 
rubber.  Fig.  272 

Rivet  together  at  short  intervals  a  strip  of  sheet  copper  and  one  of 
sheet  iron  D  (Fig.  273).  Support  this  compound  bar  so  as  to  play 
between  two  points  A  and  C,  which  are  connected  through  the  battery 
P  and  the  bell  B.  Apply  a  Bunsen  flame  to  the  bar.  It  will  warp, 
throwing  the  top  over  against  either  A  or  C,  and  will  cause  the 
bell  to  ring. 


Fig.  273 

The   experiment   shows   that   the   two  metals   expand 
unequally  and  cause  the  bar  to  warp. 


250 


HEAT 


Figure  274  illustrates  a  piece  of  apparatus  known  as  Gravesande's 
ring.     It  consists  of  a  metallic  ball  that  at  ordinary  temperatures  will 

just  pass  through  the  ring.  Heat 
the  ball  in  boiling  water.  It  will 
now  rest  on  the  ring  and  will  not  fall 
through  until  it  has  cooled. 

We  conclude  that  the  ex- 
pansion of  a  solid  takes  place 
in  every  direction. 

309.   Expansion  of  Liquids. — 

Partly  fill  several  small  air  ther- 
mometers of  the  same  capacity  with  different  liquids,  and  support  them 
vertically  in  a  metallic  vessel  (Fig.  275).  Note  the  height  of  the 
several  liquids  and  then  fill  the  vessel  with  hot  water.  The  liquids 
will  rise  in  the  tubes  but  not  equally. 

Two  facts  are  illustrated  :  first,  liquids  are  affected  by 
heat  in  the  same  way  as  solids  ;  second,  the  expansion  of 
the  liquids  is  greater  than  that  of 
the  glass  or  there  would  be  no 
apparent  increase  in  their  volume. 

Some  liquids  do  not  expand 
when  heated  at  certain  points  on 
the  thermometric  scale.  Water, 
for  example,  on  heating  from  0°  C. 
to  4°  C.  contracts,  but  above  4°  C. 
it  expands. 

310.   Expansion  of  Gases.  —  Fit  a 

bent  delivery  tube  to  a  small  Florence 

flask  (Fig.  276).     Fill  the  flask  with  air 

and  place  the  upturned  end  of  a  delivery 

tube  under  an  inverted  graduated  glass 

cylinder  filled    with  water.     Heat  the 

flask  by  immersing   it  in   a  vessel  of  Fig- 

moderately  hot  water.     The   air  will  expand  and  escape  through 

the  delivery  tube   into  the  cylinder;  note  the  amount.     Now  refill 


EXPANSION  251 

the  flask  with  some  other  gas,  as  coal  gas,  and  repeat  the  ex- 
periment. The  amount  of  gas  collected  will  be  nearly  the  same. 

Investigation  has  shown  that  all  gases  which  are  hard  to 
liquefy  expand  very  nearly  alike  at  atmospheric  pressure, 
approaching  equality  as  the 
pressure  is  diminished.  Gases 
that  are  easily  liquefied,  as  car- 
bon dioxide,  show  the  largest 
variation  in  their  expansion. 

'  311.  Coefficients  of  Expan- 
sion. —  It  appears  from  the 
preceding  experiments  that 
substances  when  heated  ex-  Fig*  276 

pand  in  every  direction.  This  expansion  in  volume  is  called 
cubical  expansion,  in  distinction  from  linear  expansion,  or 
expansion  in  length,  and  superficial  expansion,  or  expan- 
sion in  area.  The  coefficient  of  linear  expansion  is  the 
fraction  of  its  length  which  a  body  expands  when  heated 
from  0°  C.  to  1°  C. ;  the  coefficient  of  superficial  expansion 
is  the  fraction  of  its  area  which  a  body  expands  when 
heated  from  0°  C.  to  1°  C. ;  and  the  coefficient  of  cubical 
expansion  is  the  fraction  of  its  volume  which  a  body  ex- 
pands when  heated  from  0°  C.  to  1°  C.  Since  the  linear 
expansion  of  most  substances  is  found  to  be  nearly  con- 
stant for  each  degree  of  temperature,  it  is  customary  to 
determine  the  average  coefficient  for  a  change  of  several 
degrees.  If  Zx  and  Z2  represent  the  lengths  of  a  metallic 
rod  at  the  temperatures  ^  and  t2  respectively,  then 

~ — ^  =  2  .  1  is  the  expansion  for  1°,  in  which  t  is  the 
T2~h 

difference  of  temperatures.  If  a  represents  the  aver- 
age coefficient  of  expansion,  then  a  =  ~ — - ;  whence 

lit 


252  HEAT 

12  —  ^(1  -|-  at).  In  like  manner  for  volumes,  if  k  is  the 
coefficient  of  cubical  expansion,  vl  and  v2  the  volumes  at 
the  temperatures  ^  and  £2  respectively,  then 


whence  v2  =  Vj(l  4-  kt). 

In  the  case  of  solids,  superficial  and  cubical  expansion 
are  obtained  by  computation  from  the  linear  expansion, 
the  coefficient  of  the  former  being  twice  the  linear,  and 
that  of  the  latter  three  times. 

312.  Law  of  Charles.  —  It   was   shown   by   Charles,   in 
1787,  that  the  volume  of  a  given  mass  of  any  gas  under 
constant  pressure  increases  by  a  constant  fraction  of  its 
volume  at  zero  for  each  rise  of  temperature  of  1°  C.     The 
investigations  of  Regnault  and  others  show  that  the  law 
is  not  rigorously  true,  and  that  the  accuracy  of  Charles's 
law  is  about  the  same  as  that  of  Boyle's  law.     The  coeffi- 
cient of  expansion  k  of  dry  air  is  0.003665,  or  about  2T3^ 
This  fraction  may  be  considered  as  the  coefficient  of  ex- 
pansion of  any  true  gas. 

313.  The  Absolute  Scale.  —  The  law  of  Charles  leads  to  a 
fourth  scale  of  temperature  called  the  absolute  scale.     By 
this  law  the  volumes  of  any  mass  of  gas,  under  constant 
pressure,  at  0°  C.,  and  at  any  other  temperature  t°  C.,  are 
connected  by  the  following  relations  (§  311)  :  — 


At  any  other  temperature,  £',  the  volume  becomes 


EXPANSION  253 

Divide  (a)  by  (5)  and 


Suppose  now  a  new  scale  is  taken,  whose  zero  is  273 
Centigrade  divisions  below  the  freezing  point  of  water, 
and  that  temperatures  on  this  scale  are  denoted  by  T. 
Then  273  +  1  will  be  represented  by  T,  and  273  +  1'  by 
T1,  and 

v  =273  +  £    =T_ 
'  "~ 


or  the  volumes  of  the  same  mass  of  gas  under  constant  pres- 
sure are  proportional  to  the  temperatures  on  this  new  scale. 
The  point  273°  below  0°  C.  is  called  the  absolute  zero,  and 
the  temperatures  on  this  scale,  absolute  temperatures.  Up 
to  the  present  it  has  not  been  found  possible  to  cool  a 
body  to  the  absolute  zero  ;  but  by  evaporating  liquid 
hydrogen  under  very  low  pressure,  a  temperature  esti- 
mated to  be  within  9°  of  the  absolute  zero  has  been  ob- 
tained by  Professor  Dewar;  and  Professor  Onnes,  by 
liquefying  helium,  believes  that  he  obtained  a  tempera- 
ture within  3°  of  the  absolute  zero. 

314.  The  Laws  of  Boyle  and  Charles  Combined.  —  If  v,  p, 
and  T  denote  the  volume,  pressure,  and  absolute  tempera- 
ture of  a  given  mass  of  gas,  then  by  Boyle's  law  (§  74) 

v  oc  -,  when  T  is  constant  ;    and  by  the  law  of  Charles, 

P 

v  oc  T,  when  p  is  constant.  Therefore  when  T  and  p 
both  vary,  v  varies  directly  as  T  and  inversely  as  jp,  or 

T 

v  oc  —  .     Whence  pv  oc  T,  or  pv  =  constant  x  T.     This 

P 


254  HEAT 

relation  is  known  as  the  "  gas  equation "  and  is  written 

pv  =  RT.     .     .     .  (Equation  33) 

R  is  the  constant  which  converts  a  proportionality  into 
an  equality.  It  follows  that  not  only  is  the  volume  of  a 
given  mass  of  gas  under  constant  pressure  proportional  to 
its  absolute  temperature  (§  313),  but  the  product  of  the 
pressure  and  volume  of  a  given  mass  of  gas  is  proportional 
to  its  absolute  temperature. 

To  illustrate  the  use  of  the  above  relation  :  If  20  cm.8 
of  gas  at  20°  C.  is  under  a  pressure  of  76  cm.  of  mercury, 
what  will  be  the  pressure  when  its  volume  is  30  cm.3  and 
temperature  50°  C.  ? 

From  equation  (33),    ^  is  a  constant, 
pv      pfvr 

~T=2~Tr' 

Hence  76  x  20  _     p  x  30 

273+20~273  +  50' 
from  which  p  =  55.85  cm. 

315.  Force  of  Contraction  and  Expansion.  —  Fill  a  small  test 
tube  about  one  quarter  full  of  water,  and  close  the  end  by  fusion. 
Lay  it  in  an  empty  sand  bath  on  the  ring  of  an  iron  stand.  Apply 
heat,  and  stand  at  a  safe  distance.  In  a  few  minutes  there  will  be 
a  loud  report,  caused  by  the  bursting  of  the  tube. 

The  force  of  expansion  or  of  contraction  of  a  subtance 
is  evidently  equal  to  the  force  necessary  to  compress  or 
expand  it  to  the  same  extent  by  mechanical  means,  and 
hence  can  be  computed  by  proceeding  in  the  manner  illus- 
trated in  the  following  example :  A  bar  of  malleable  iron, 
one  square  inch  in  cross-sectional  area,  if  placed  under 
the  tension  of  a  ton,  increases  in  length  0.0001  of  itself. 


EXPANSION  255 

The  coefficient  of  linear  expansion  of  iron  is  0.0000122. 
Since  0.0001  -r-  0.000122  =  8  +  ,  a  change  of  temperature 
of  about  8°  C.  will  produce  the  same  change  in  the  length 
of  the  bar  as  a  force  of  one  ton. 

It  takes  a  pressure  of  600  atmospheres  to  keep  mercury 
from  expanding  when  heated  from  0°  C.  to  10°  C. 

316.  Applications  of  Expansion  and  Contraction.  —  Many 
familar  phenomena  are  accounted  for  by  expansion  or  con- 
traction attending  changes  of  temperature.  If  hot  water 
is  poured  into  a  thick  glass  tumbler,  the  glass  will  prob- 
ably break  because  of  the  stress  produced  by  the  sudden 
expansion  of  its  inner  surface.  The  principle  of  unequal 
expansion  is  employed  in  thermometers,  in  the  compen- 
sated clock  pendulum  and  in  the  bal- 
ance wheel  of  a  watch  (Fig.  277), 
in  which  the  rim  is  made  in  two  sec- 
tions, each  composed  of  two  metals 
soldered  together  side  by  side,  with 
the  more  expansible  metal  on  the  out- 
side. When  the  temperature  rises,  the 
ends  #,  a1  move  inward.  Glass  and  Fig  277 

platinum  have  nearly  the  same  coef- 
ficient of  expansion.  For  that  reason  platinum  is  in  great 
demand  in  the  manufacture  of  incandescent  electric  lamps, 
since  it  does  not  crack  the  glass  when  it  cools.  Iron  tires 
are  fitted  to  wheels  and  then  expanded  by  heating  so  that 
they  slip  on  easily ;  on  cooling,  they  contract  and  com- 
press the  wheel.  The  rivets  which  hold  together  the 
plates  of  steam  boilers  are  inserted  red-hot,  and  hammered 
down.  The  contracting  rivets  press  the  plates  together 
with  great  force.  In  all  heavy  iron  structures,  such  as 
railroad  bridges,  a  certain  freedom  of  motion  of  the  parts 


256  HEAT 

must  be  provided  for  ;  otherwise,  the  changes  in  length 
attending  variations  in  temperature  would  have  a  disas- 
trous effect.  Sidewalks  of  artificial  stone  should  have 
spaces  left  for  expansion  to  prevent  "  buckling. "  Crys- 
talline rocks,  on  account  of  unequal  expansion  in  different 
directions,  are  slowly  disintegrated  by  changes  of  tem- 
perature ;  and  for  the  same  reason  quartz  crystals,  when 
strongly  heated,  fly  in  pieces. 


Questions  and  Problems 

1.  What  is  the  objection  in  stringing  telegraph  wires  in  the  sum- 
mer time  to  stretching  them  tight  ? 

2.  Why  will  warming  the  neck  of  a  bottle  often  loosen  a  glass 
stopper  that  has  stuck  ? 

3.  A  quantity  of  alcohol  that  measures  20  gallons  on  thfc  first  of 
January  might  measure  as  much  as  21  gallons  on  the  first  of  July. 
Explain. 

4.  Set  a  pan  even  full  of  cold  water  on  a  hot  stove.     In  a  short 
time  it  will  begin  to  overflow.     Why  ? 

5.  Fill  a  vessel  containing  a  piece  of  ice  level  full  of  water.    When 
the  ice  melts  the  water  level  neither  rises  nor  falls.     Why  ? 

6.  An  iron  rod  60  cm.  long  at  20°  C.  was  60.055  cm.  long  at  95°  C. 
Calculate  the  coefficient  of  expansion. 

7.  A  brass  rod  was  100  cm.  long  at  20°  C.     What  will  be  its  length 
at  0°  C.,  if  the  coefficient  of  linear  expansion  is  0.0000186  ? 

8.  A  copper  bar  62.5  cm.  long  at  5°C.  expands  by  1.1  mm.  when 
heated  to  98°  C.     Find  its  coefficient  of  expansion. 

9.  An  iron  bridge  is  300  ft.  long.      Calculate  the  variation  in 
length  it  will  undergo  between  the  temperatures  —  10°  C.  and  40°  C., 
the  coefficient  of  expansion  of  iron  being  0.0000122. 

10.  A  glass  graduate  holds  one  liter  at  15°  C.     How  much  will  it 
hold  at  25°  C.,  if  the  coefficient  of  cubical  expansion  is  0.000025  ? 

11.  Two  metal  bars  are  each  one  meter  long  at  10°  C.      How 
much  will  they  differ  in  length  at  40°  C.  if  one  is  steel  (coefficient  of 


MEASUREMENT  OF  HEAT  257 

linear  expansion,   0.0000132)  and  the  other  aluminum  (coefficient, 
0.0000222)  ? 

12.  If  1500  cm. 8  of  air  at  15°  C.  be  changed  in  temperature  to  50°  C., 
what  will  the  volume  be  if  the  pressure  is  constant? 

13.  In  an  experiment  to  determine  the  coefficient  of  expansion  of 
air  the  following  data  were  obtained  :  Volume  of  air  at  0  °  C.,  145 
cm.8;  at  100° C.,  198  cm.3,  the  pressure  remaining  the  same.     Calcu- 
late the  coefficient  of  expansion. 

14.  A  flask  filled  with  air  at  20°  C.  and  under  75  cm.  pressure  is 
stoppered  and  heated  to  70°  C.     Assuming  that  the  flask  does  not  ex- 
pand appreciably,  under  what  pressure  will  the  air  be? 

15.  If  a  liter  of  dry  air  at  0°  C.  weighs  1.3  gm.,  how  many  liters 
will  weigh  10  gm.  if  the  pressure  be  reduced  to  one  half  and  the  tem- 
perature be  raised  to  100°  C.  ? 

• 
IV.    MEASUREMENT   OF  HEAT 

317.  The  Unit  of  Heat.  —  The  unit  of  heat  in  the  e.g.  s. 
system  is  the  calorie.     It  is  defined  as  the  quantity  of  heat 
that  will  raise  the  temperature  of  one  gram  of  water  one  de- 
gree Centigrade.     There  is  no  agreement  as  to  the  position 
of  the  one  degree  on  the  thermometric  scale,  although  it  is 
known  that  the  unit  quantity  of  heat  varies  slightly  at 
different    points  on   the  scale.       If   the  degree   interval 
chosen  is  from  15°  to  16°C.,  the  calorie  is  then  the  one 
hundredth  part  of  the  heat  required  to  raise  the  tempera- 
ture of  one  gram  of  water  from  0°  to  100°  C. 

In  engineering  practice  in  England  and  America  the 
British  thermal  unit  (B.  T.  U.)  is  commonly  employed.  It 
is  the  heat  required  to  raise  the  temperature  of  one  pound 
of  water  one  degree  Fahrenheit. 

318.  Thermal  Capacity.  —  The  thermal  capacity  of  a  body 
is  the  number  of  calories  required  to  raise  its  temperature 
one  degree  Centigrade.     The  thermal  capacity  of  equal 


258  HEAT 

masses  of  different  substances  differs  widely.  For  example, 
if  100  gm.  of  water  at  0°  C.  be  mixed'  with  100  gm.  at 
100°  C.,  the  temperature  of  the  whole  will  be  very  nearly 
50°  C.  But  if  100  gm.  of  copper  at  100°  C.  be  cooled  in 
100  gm.  of  water  at  0°C.,  the  final  temperature  will  be 
about  9.1°C.  The  heat  lost  by  the  copper  in  cooling 
through  90.9°  is  sufficient  to  heat  the  same  mass  of  water 
only  9.1°,  that  is,  the  thermal  capacity  of  water  is  about 
ten  times  as  great  as  that  of  an  equal  mass  of  copper. 

319.  Specific  Heat.  —  The  specific  heat  of  a  substance  is 
the  number  of  calories  of  heat  required  to  raise  the  tem- 
perature of  one  gram  of  it  through  one  degree  Centigrade. 
It  may  be  defined  independently  of  any  temperature  scale 
as  the  ratio  between  the  number  of  units  of  heat  required 
to  raise  the  temperature  of  equal  masses  of  the  substance 
and  of  water  through  one  degree.     The  specific  heat  of 
mercury  is  .033,  that  is,  the  heat  that  will  raise  1  gm.  of 
mercury  through  1°  C.  will  raise  1  gm.  of  water  through 
only  0.033°  C. 

320.  Numerical  Problem  in  Specific  Heat.  —  The  principle  ap- 
plied in  the  solution  of  such  problems  is  that  the  gain  or  loss  of  heat 
by  the  water  is  equal  to  the  loss  or  gain  of  heat  by  the  body  introduced 
into  the  water.     The  gain  or  loss  of  heat  by  the  body  is  equal  to  the 
product  of  its  mass,  its  specific  heat,  and  its  change  of  temperature. 

To  illustrate :  20  gm.  of  iron  at  98°  C.  are  placed  in  75  gm.  of  water 
at  10°  C.  contained  in  a  copper  beaker  weighing  15  gm.,  specific  heat 
0.095.  The  resulting  temperature  of  the  water  and  the  iron  is  12.5°  C. 
Find  the  specific  heat  of  iron. 

The  thermal  capacity  of  the  beaker  is  15  x  0.095  =  1.425  calories. 
The  heat  lost  by  the  iron  is  20  x  s  x  (98  —  12.5)  calories,  in  which  s  rep- 
resents the  specific  heat  of  iron,  and  (98  —  12.5)  its  change  of  temper- 
ature. The  heat  gained  by  the  water  and  the  copper  vessel  is  (75  + 
1.425)  x  (12.5  —  10)  calories;  the  second  factor  is  the  gain  in  temper- 


QUESTIONS  AND  PROBLEMS  259 

ature  of  the  water  and  the  beaker.  It  follows  by  equating  these 
two  quantities  that  20  x  s  x  (98  -  12.5)  =  (75  +  1.425)  x  (12.5  -  10). 
Solving  for  s,  we  have  s  =  0.112  calorie  per  gram. 


Questions  and  Problems 

1.  If  water  were  used  as  the  substance  in  a  thermometer,  what 
would  be  the  lowest  temperature  it  would  register  ? 

2.  Give  three  reasons  why  mercury  is  a  more  suitable  substance 
for  thermometers  than  water. 

3.  If  a  pound  of  water  and  a  pound  of  iron  expose  the  same  sur- 
face area  to  the  direct  rays  of  the  sun,  which  will  show  the  greater 
change  of  temperature  in  an  hour? 

4.  If  equal  quantities  of  heat  are  applied  to  equal  masses  of  iron 
and  copper,  which  will  show  the  greater  change  of  temperature  ? 

5.  Which  would  be  the  more  efficient  foot  warmer,  a  rubber  bag 
containing  5  Ib.  of  water  at  80°  C.,  or  a  5-lb.  block  of  iron  also  at 
80°  C.  ?     Give  reasons. 

6.  If  a  number  of  balls  of  the  same  mass  but  of  different  materials 
are  heated  in  boiling  water  and  are  then  placed  on  a  cake  of  wax,  will 
they  all  melt  the  same  quantity  of  wax  ?    Why  ? 

7.  How  many  calories  of  heat  will  it  take  to  raise  the  temperature 
of  50  gm.  of  water  from  10°  C.  to  70°  C.?     If  this  heat  were  all  applied 
to  1  liter  of  water  at  2C°  C.,  to  what  temperature  would  it  raise  the 
water  ? 

8.  If  90  gm.   of  mercury  at  100°  C.  are  stirred  with  100  gm.  of 
water  at  20°  C.  and  the  resulting  temperature  is  22.3°  C.,  what  is  the 
specific  heat  of  mercury  ? 

9.  If  the  specific  heat  of  iron  is  0.112,  how  much  heat  will  be  re- 
quired to  raise  the  temperature  of  2.5  kgm.  of  iron  from  15°  C.  to 
100°  C.? 

10.  A  copper  ball  weighing  5  kgm.  is  heated  to  a  temperature  of 
100°  C.,  and  when  placed  in  water  raises  its  temperature  from  20°  C. 
to  25°  C.  How  many  grams  of  water  are  there,  the  specific  heat  of 
copper  being  0.095? 


260  HEAT 

V.    CHANGE  OF  STATE 

321.  The  Melting  Point.  —  A  body  is  said  to  melt  or  fuse 
when  it  changes  from  the  solid  to  the  liquid  state  by  the 
application  of  heat.     The  change  is  called  melting,  fusion, 
or  liquefaction.     The  temperature  at  which  fusion  takes 
place  is  called  the  melting  point.     Solidification  or  freezing 
is  the  converse  of  fusion,  and  the  temperature  of  solidifi- 
cation is  usually  the  same  as   the  melting  point   of   the 
same  substance.     Water,  if  undisturbed,  may  be  cooled  a 
number  of  degrees  below  0°  C.,  but  if  it  is  disturbed  it 
usually  freezes  at  once,  and  its  temperature  rises  to  the 
freezing  point. 

The  melting  point  of  crystalline  bodies  is  well  marked. 
A  mixture  of  ice  and  water  will  remain  without  change  if 
the  temperature  of  the  room  is  0°  C.  ;  but  if  the  tempera- 
ture is  above  zero,  some  of  the  ice  will  melt  ;  if  it  is  below 
zero,  some  of  the  water  will  freeze.  Some  substances,  like 
wax,  glass,  and  wrought  iron,  have  no  sharply  defined 
melting  point.  They  first  soften  and  then  pass  more  or 
less  slowly  into  the  condition  of  a  viscous  liquid.  It  is 
this  property  which  permits  of  the  bending  and  molding 
of  glass,  and  the  welding  and  forging  of  iron. 

322.  Change  in  Volume  accompanying  Fusion. — Fit  to  a 

small  bottle  a  perforated  stopper  through  which  passes  a  fine  glass 
tube.  Fill  with  water  recently  boiled  to  expel  the  air,  the  water  ex- 
tending halfway  up  the  tube.  Pack  the  apparatus  in  a  mixture  of 
salt  and  finely  broken  ice.  The  water  column  at  first  will  fall  slowly, 
but  in  a  few  minutes  it  will  begin  to  rise,  and  will  continue  to  do  so 
until  water  flows  out  of  the  top  of  the  tube.  The  water  in  the  bottle 
freezes,  expands,  and  causes  the  overflow. 

Most  substances  occupy  a  larger  volume  in  the  liquid 
state  than  in  the  solid  ;  that  is,  they  expand  in  liquefying. 


CHANGE  OF  STATE  261 

A  few  substances,  like  water  and  bismuth,  expand  in 
solidifying.  When  water  freezes,  its  volume  increases 
9  per  cent.  If  this  expansion  is  resisted,  water  in  freezing 
is  capable  of  exerting  a  force  of  about  2000  kgm.  per 
square  centimeter. 

323.  Effect  of  Pressure  on  the  Melting  Point.  —  Support  a 
rectangular  block  or  prism  of  ice  on  a  stout  bar  of  wood.     Pass  a 
thin  iron  wire  around  the  ice  and  the  bar  of  wood,  and  suspend  on  it 
a  weight  of  about  25  Ib.     The  pressure  of  the  wire  lowers  the  melting 
point  of  the  ice  immediately  under  it  and  the  ice  melts ;  the  water, 
after  passing  around  the  wire,  where  it  is  relieved  of  pressure,  again 
freezes.     In  this  way  the  wire  passes  slowly  through  the  ice,  leaving 
the  block  solidly  frozen. 

A  rough  numerical  statement  of  the  effect  of  pressure 
on  the  freezing  point  of  water  is  that  a  pressure  of  one 
ton  per  square  inch  lowers  the  freezing  point  to  —  1°  C. 
Familiar  examples  of  refreezing,  or  regelation,  are  the 
hardening  of  snowballs  under  the  pressure  of  the  hands, 
the  formation  of  solid  ice  in  a  roadway  where  it  is 
compressed  by  vehicles  and  the  hoofs  of  horses,  and 
frozen  footforms  in  compact  ice  after  the  loose  snow  has 
melted  around  them.  The  ice  of  a  glacier  melts  where  it 
is  under  the  enormous  pressure  of  the  descending  masses 
above  it.  The  melting  permits  the  ice  to  accommodate 
itself  to  abrupt  changes  in  the  rocky  channel,  and  a  slow 
iceflow  results.  As  soon  as  the  pressure  at  any  surface 
is  relieved,  the  water  again  freezes,  "fr  <?&£&/-&. 

jf  rr&/-f7x4 

324.  Heat  of  Fusion. — When  a  solid  melts,  a  quantity 
of  heat  disappears  ;  and,  conversely,  when  a  liquid  solidi- 
fies, the   amount  of   heat  generated   is   the  same  as  dis- 
appears  during  liquefaction.     The   heat   of  fusion   of   a 
substance  is  the  number  of   calories   required  to  melt  a 


262  HEAT 

gram  of  it  without  change  of  temperature.     The  heat  of 
fusion  of  ice  is  80  calories. 

As  an  illustration  of  the  heat  of  fusion,  place  200  gm.  of  clean  ice, 
broken  into  small  pieces,  into  500  gm.  of  water  at  60°  C.  When  the 
ice  has  melted,  the  temperature  will  be  about  20°  C.  The  heat  lost 
by  the  500  gm.  of  water  equals  500  x  (60  -  20)  =20,000  calories. 
This  heat  goes  to  melt  the  ice  and  to  raise  the  water  from  it  from 
0°  C.  to  20°  C.  The  latter  is  200  x  20  =  4000  calories.  The  re- 
mainder, 20,000—  4000  =  16,000  calories,  went  to  melt  the  ice.  Then 
the  heat  of  fusion  of  ice  is  16,000  -f-  200  =  80  calories  per  gram, 

325.  Heat  lost  in  Solution.  — Fill  a  glass  beaker  partly  full  of 
water  at  the  temperature  of  the  room,  and  add  some  ammonium 
nitrate  crystals.     The  temperature  of  the  water  will  fall  as  the  crys- 
tals dissolve. 

This  experiment  illustrates  the  fact  that  heat  disappears 
when  a  body  passes  from  the  solid  to  the  liquid  state  by 
solution.  The  use  of  salt  in  soup  or  of  sugar  in  tea  absorbs 
heat.  The  heat  energy  is  used  to  pull  down  the  solid 
structure. 

326.  Freezing  Mixtures.  —  Freezing  mixtures  are  based 
on  the  absorption  of  heat  necessary  to  give  fluidity.     Salt 
water  freezes  at  a  lower  temperature  than   fresh   water. 
When  salt  and  snow  or  pounded  ice  are  mixed  together, 
both  become  fluid  and  absorb  heat  in  the  passage  from  the 
one  state  to  the  other.     By  this  mixture  a  temperature  of 
—  22°  C.  may  be  obtained.     Still  lower  temperatures  may 
be   reached   with   other   mixtures,   notably  with  sulpho- 
cyanide  of  sodium  and  water. 

327.  Vaporization.  —  Pour  a  few  drops  of  ether  into  a  beaker  and 
cover  loosely  with  a  plate  of  glass.     After  a  few  seconds  bring  a  lighted 
taper  to  the  mouth  of  the  beaker.     A  sudden  flash  will  show  that  the 
vapor  of  ether  was  mixed  with  the  air. 

Support  on  an  iron  stand  a  Florence  flask  two  thirds  full  of  water 


CHANGE  OF  STATE  263 

and  apply  heat.  In  a  short  time  bubbles  of  steam  will  form  at  the 
bottom  of  the  flask,  rise  through  the  water,  and  burst  at  the  top,  pro- 
ducing violent  agitation  throughout  the  mass. 

Vaporization  is  the  conversion  of  a  substance  into  the 
gaseous  form.  If  the  change  takes  place  slowly  from  the 
surface  of  a  liquid,  it  is  called  evaporation;  but  if  the  liquid 
is  visibly  agitated  by  rapid  internal  evaporation,  the  process 
is  called  ebullition  or  boiling. 

328.  Sublimation. — When  a  substance  passes   directly 
from  the  solid  to  the  gaseous  form  without  passing  through 
the  intermediate  state  of  a  liquid,  it  is  said  to  sublime. 
Arsenic,  camphor,  and  iodine  sublime  at  atmospheric  pres- 
sure, but  if  the  pressure  be  sufficiently  increased,  they  may 
be  fused.     Ice  also  evaporates  slowly  even  at  a  temperature 
below  freezing.     Frozen  clothes  dry  in  the  air  in  freezing 
weather.     At  a,  pressure  less  than  4.6  mm.  of  mercury,  ice 
is  converted  into  vapor  by  heat  without  melting. 

329.  The  Spheroidal  State.  —  When  a  small  quantity  of 
liquid  is  placed  on  hot  metal,  as  water  on  a  red-hot  stove, 
it  assumes  a  globular  or  spheroidal  form,  and  evaporates 
at  a  rate  between  ordinary  evaporation  and  boiling.     It 
is  then  in  the  spheroidal  state.      The  vapor  acts  like  a 
cushion  and  prevents  actual  contact  between  the  liquid 
and  the  metal.     The  globular  form  is  due  to  surface  ten- 
sion.    Liquid  oxygen  at  —  180°  C.  assumes  the  spheroidal 
form  on  water.     The  temperature  of  the  water"  is  relatively 
high  compared  with  that  of  the  liquid  oxygen. 

330.  Cold  by  Evaporation.  —  Tie  a  piece  of  fine  linen  around  the 
bulb  of  a  thermometer  and  pour  on  it  a  few  drops  of  sulphuric  ether. 
The  temperature  will  at  once  begin  to  fall,  showing  that  the  bulb  has 
been  cooled. 


264 


HEAT 


In  the  evaporation  of  ether,  some  of  the  heat  of  the 
thermometer  is  used  to  do  work  on  the  liquid.  The  rapid 
evaporation  of  liquid  ammonia  is  utilized  in  making  arti- 
ficial ice.  Large  tanks  of  salt  brine  are  cooled  by  coils  of 
pipe  containing  liquid  ammonia.  A  pump  lowers  the  pres- 
sure in  these  coils  and  the  ammonia  evaporates  rapidly  with 
the  production  of  a  low  temperature.  Cans  of  water  set 
in  these  tanks  are  frozen.  The  ammonia  gas  is  afterwards 
reduced  again  to  the  liquid  form  by  pressure  and  cooling. 
Sprinkling  the  floor  of  a  room  cools  the  air,  because  of 
the  heat  expended  in  evaporating  the  water.  Porous 
water  vessels  keep  the  water  cool  by  the  evaporation  of  the 
water  from  the  outside  surface.  Liquid  carbon  dioxide 
is  readily  frozen  by  its  own  rapid  evaporation.  Dewar 
liquefied  oxygen  by  means  of  the  temperature  obtained 
through  the  successive  evaporation  of  liquid  nitrous  oxide 
and  ethylene.  Similarly,  by  the  evaporation  of  liquid  air 
he  has  liquefied  hydrogen.  The  evaporation  of  liquid 
hydrogen  under  reduced  pressure 
has  enabled  him  to  obtain  a  temper- 
ature but  little  removed  from  the 
absolute  zero,  -  273°  C. 

331.  Effect  of  Pressure  on  the  Boil- 
ing Point.  —  Place  a  flask  of  warm  water 
under  the  receiver  of  an  air  pump.  It  will 
boil  violently  when  the  receiver  is  ex- 
hausted. 

Fill   a  round-bottomed    Florence   flask 
half  full   of  water  and   heat  till  it  boils 
vigorously.     Cork    the    flask,   invert,   and 
PJ~  278  support  it  on  a  ring  stand  (Fig.  278).   The 

boiling  ceases,  but  is  renewed  by  applying 

cold  water  to  the  flask.     The  cold  water  condenses  the  vapor,  and  re- 
duces the  pressure  within  the  flask  so  that  the  boiling  begins  again. 


CHANGE  OF  STATE 


265 


The  effect  of  pressure  on  the  boiling  point  is  seen  in 
the  low  temperature  of  boiling  water  at  high  elevations, 
and  in  the  high  temperature  of  the  water  under  pressure 
in  digesters  used  for  extracting  gelatine  from  bones.  The 
boiling  point  of  water  falls  1°  C.  for  an  increase  in  elevation 
of  about  295  m.  At  Quito  the  boiling  point  is  near  90°  C. 

332.  Heat  of  Vaporization.  —  The  heat  of  vaporization  is 
the  number  of  calories  required  to  change  one  gram  of 
a  liquid  at  its  boiling  point  into  vapor  at  the  same  tem- 
perature. Water  has  the  greatest  heat  of  vaporization  of 
all  liquids.  The  most  carefully  conducted  experiments 
show  that  the  heat  of  vaporization  of  water  under  a  pressure 
of  one  atmosphere  is  536.6  calories  per  gram. 

Set  up  apparatus  like  that  shown  in  Fig.  279.  The  steam  from  the 
boiling  water  is  conveyed  into  a  beaker  containing  a  known  quantity 
of  water  at  a  known  temperature. 
The  increase  in  the  mass  of  the  water 
gives  the  amount  of  steam  con- 
densed. The  "  trap  "  in  the  delivery 
tube  catches  the  water  that  con- 
denses before  it  reaches  the  beaker. 
Suppose  that  the  experiment  gave 
the  following  data :  Amount  of 
water  in  the  beaker,  400  gm.  at  the 
beginning,  414.1  gm.  at  the  end,  in- 
cluding the  thermal  capacity  of  the 
beaker  in  terms  of  water ;  tempera- 
ture at  the  beginning,  20°  C.,  and 
at  the  end,  41°  C. ;  observed  boil- 
ing point,  99°  C. ;  there  were  14.1 
gm.  of  steam  condensed.  Now,  by 
the  principle  that  the  heat  lost 
or  given  off  by  the  steam  equals  that  gained  by  the  water,  we  have 

400  x  (41  -  20)  =  14.1  x  I  +  14.1  x  (99  -  41)  ; 
whence  /  =  537.7  cal.  per  gram. 


Fig.  279 


266  HEAT 

333.  The  Dew  Point.  —  The  dew  point  is  the  temperature 
at  which  the  aqueous  vapor  of  the  atmosphere  begins  to  con- 
dense. If  water  at  the  temperature  of  the  room  be  poured 
into  a  polished  nickel-plated  beaker  and  some  small  pieces 
of  ice  be  added  with  stirring,  a  mist  will  soon  collect  on 
the  outside  of  the  vessel.  The  temperature  of  the  water 
is  then  the  dew  point.  The  formation  of  clouds,  the  pre- 
cipitation of  dew,  and  the  "  sweating  "  of  pitchers  containing 
ice  water  are  evidence  of  the  existence  of  water  vapor  in 
the  atmosphere.  Dew  collects  on  objects  when  their  tem- 
perature drops  below  the  dew  point. 

The  amount  of  moisture  that  the  air  can  contain  depends 
on  the  temperature.  The  terms  dryness  and  moistness, 
applied  to  the  air,  are  purely  relative.  They  indicate  the 
proportion  of  water  vapor  actually  present  in  comparison 
with  what  the  air  could  hold  when  saturated  at  the  same 
temperature.  The  air  is  saturated  at  the  dew  point.  A 
dry  day  is  one  on  which  the  dew  point  is  much  below  the 
temperature  of  the  air ;  a  damp  day  is  one  on  which  the 
dew  point  is  close  to  the  temperature  of  the  air. 

Questions  and  Problems 

1.  Why  does  a  drop  or  two  of  alcohol  feel  cold  on  the  hand  ? 

2.  Why  do  the  windows  of  an  occupied  house  frost  over  on  the 
inside  on  a  very  cold  day  ? 

3.  Why  is  the  face  cooled  by  fanning  ? 

4.  Can  water  be  heated  above  100°  C.  ?    How  ? 

5.  Why  does  warming  a  room  make  it  drier? 

6.  Why  does  one  feel  cold  when  sitting  in  a  draft? 

7.  Why  is  Water  better  than  any  other  liquid  for  heating  pur- 


8.  Why  is  there  no  dew  on  windy  nights  ? 

9.  Why  will  pressure  cause  two  blocks  of  ice  to  adhere?     Will 
they  adhere  if  their  temperature  is  much  below  freezing  ? 


CHANGE  OF  STATE  267 

10.  How  much  heat  does  it  take  to  melt  100  gm.  of  ice? 

11.  How  much  heat  does  it  take  to  convert  100  gm.  of  water  at 
100°  C.  into  steam  at  100°  C.? 

12.  How  much  heat  will  it  take  to  convert  75  gm.  of  ice  into 
steam? 

13.  50  gm.  of  ice  at  0°  C.  are  put  into  50  gm.  of  water  at  40°  C. 
How  much  of  the  ice  will  melt  ? 

14.  How  much  ice  must  be  put  into  100  gm.  of  water  at  60°  C.  to 
lower  the  temperature  to  20°  C.  ? 

15.  If  25  gm.   of  steam  at  100°  C.  are  condensed  in  20  gm.  of 
water  at  15°  C.,  wnat  will  be  the  resulting  temperature  ? 

16.  How  much  ice  can  be  melted  by  50  gm.  of  steam  at  100°  C.  if 
none  of  the  heat  is  lost  ? 


VI.    TRANSMISSION  OF  HEAT 

334.  Conduction.  —  Twist  together  two  stout  wires,  iron  and  cop- 
per, of  the  same  diameter,  forming  a  fork  with  long  parallel  prongs 
and  a  short  stem.  Support  them  on  a  wire  stand  (Fig.  280),  and  heat 
the  twisted  ends.  After  several  minutes  find  the  point  on  each  wire, 


Fig.  280 

farthest  from  the  flame,  where  a  sulphur  match  ignites  when  held 
against  the  wire.  f  This  point  will  be  found  farther  along  on  the  cop- 
per than  on  the  iron,  showing  that  the  former  has  led  the  heat  farther 
from  its  source. 

Prepare  a  cylinder  of  uniform  diameter,  half  of  which  is  made  of 
brass  and  half  of  wood.     Hold  a  piece  of  writing  paper  firmly  around 


268 


HEAT 


the  junction  like  a  loop  (Fig.  281).     By  applying  a  Bunsen  flame  the 
paper  in  contact  with  the  wood  is  soon  scorched,  while  the  part  in 

contact  with  the  brass  is  scarcely  in- 
jured. The  metal  conducts  the  heat 
away  and  keeps  the  temperature  of  the 
paper  below  the  point  of  ignition.  The 
wood  is  a  poor  conductor. 


These  experiments  show  that 
solids  differ  in  their  conductivity 
for  heat.  The  metals  are  the  best 
conductors ;  wood, leather,  flannel, 
and  organic  substances  in  general 

are  poor  conductors ;  so  also  are  all  bodies  in  a  powdered 
state,  owing  doubtless  to  a  lack  of  continuity  in  the  material. 

335.  Conductivity  of  Liquids.  —  Pass  a  glass  tube  surmounted 
with  a  bulb  through  a  cork  fitted  to  the  neck  of  a  large  funnel.     Sup- 
port the  apparatus  as  shown  in  Fig.  282. 

The  glass  stem  should  stand  in  colored 
water.  Heat  the  bulb  slightly  to  expel 
some  air,  so  that  the  liquid  will  rise  in  the 
tube.  Fill  the  funnel  with  water,  covering 
the  bulb  to  the  depth  of  about  one  centi- 
meter. Pour  a  spoonful  of  ether  on  the 
water  and  set  it  on  fire.  The  steadiness 
of  the  index  shows  that  little  if  any  of  the 
heat  due  to  the  burning  ether  is  conducted 
to  the  bulb. 

This  experiment  shows  that 
water  is  a  poor  conductor  of  heat. 
This  is  equally  true  of  all  liquids 
except  molten  metals. 

336.  Conductivity  of  Gases.  —  The 

conductivity  of  gases  is  very  small, 

and  its  determination  is  very  difficult  because  of  radia- 
tion and  convection.     The  conductivity  of  hydrogen   is 


TRANSMISSION  OF  HEAT  269 

about  7.1  times  that  of  air,  while   the  conductivity  of 
water  is  25  times  as  great. 

337.  Applications  of  Conductivity.  —  If  we  touch  a  piece 
of  marble  or  iron  in  a  room,  it  feels  cold,  while  cloth  and 
wood  feel  distinctly  warmer.  The  explanation  is  that 
the  articles  which  feel  cold  are  good  conductors  of  heat 
and  carry  it  away  from  the  hand,  while  the  poor  con- 
ductors do  not. 

The  good  heat-conducting  property  of  copper  or  brass 
is  turned  to  practical  account  in  the  Davy  miner's  lamp. 
The  flame  is  completely  inclosed  in  metal 
and  fine  wire  gauze.  The  gauze  by  con- 
ducting away  heat  keeps  any  fire  damp 
outside  the  lamp  below  the  temperature  of 
ignition  and  so  prevents  explosions.  The 
action  of  the  gauze  is  readily  illustrated 
by  holding  it  over  the  flame  of  a  Bunsen 
burner  (Fig.  283).  The  flame  does  not 
pass  through  unless  the  gauze  is  heated 
to  redness.  If  the  gas  is  first  allowed 
to  stream  through  the  gauze,  it  may  be 
lighted  on  top  without  being  ignited  below. 

The  handles  on  metal  instruments  that 
are  to  be  heated  are  usually  made  of  some  poor  conductor, 
as  wood,  bone,  etc.  ;  or  else  they  are  insulated  by  the  in- 
sertion of  some  non-conductor,  as  in  the  case  of  the 
handles  to  silver  teapots,  where  pieces  of  ivory  are  inserted 
to  keep  them  from  becoming  too  hot. 

The  non-conducting  character  of  air  is  utilized  in  houses 
with  hollow  walls,  in  double  doors  and  double  windows, 
and  in  clothing  of  loose  texture.  The  warmth  of  woolen 
articles  and  of  fur  is  due  mainly  to  the  fact  that  much 


270 


HEAT 


air   is   inclosed    within   them   on  account  of  their  loose 
structure. 

338.  Convection.  —  Set  up  apparatus  as  shown  in  Fig.  284,  and 
support  it  on  a  heavy  iron  stand.     Fill  the  flask  and  connecting  tubes 

with  water  up  to  a  point  a  little  above  the  open  end 
of  the  vertical  tube  at  C.  Apply  a  Bunseri  flame  to 
the  flask  B.  A  circulation  of  water  is  set  up  in  the 
apparatus,  as  shown  by  the  arrows.  The  circulation 
is  made  visible  by  coloring  the  water  in  the  reservoir 
blue  and  that  in  the  flask  red. 

The  process  of  conveying  heat  by  the 
transference  of  the  heated  matter  itself  is 
known  as  convection.  Currents  set  up  in  this 
manner  are  called  convection  currents.  The 
heating  of  buildings  by  hot  water  conveyed 
though  a  pipe  to  an  expansion  tank  at  the 
top  of  the  building  and  back  through  radi- 
Fig.  284  ators  in  the  rooms  is  an  application  of 
convection.  The  circulation  is 
mantained  because  the  column 
of  hot  water  leading  to  the  ex- 
pansion tank  is  hotter  and  there- 
fore lighter  then  the  water  in 
the  return  pipes. 

339.  Convection  in  Gases.  —  Set 

a  short  piece  of  lighted  candle  in  a* 

shallow  beaker   and   place  over  it  a 

lamp  chimney.     Pour  into  the  beaker 

enough  water  to  close  the  lower  end 

of  the  chimney.     Place  in  the  top  of 

the  chimney   a  T-shaped  piece  of  tin 

as   a  short  partition   (Fig.  285).     If 

a  piece  of  smoldering  paper  be  held  over  one  edge  of  the  chimney, 

the  smoke  will  pass  down  one  side  of  the  partition  and  up  the  other. 

If  the  partition  be  removed,  the  flame  will  usually  go  out. 


Fig.  285 


TRANSMISSION  OF  HEAT  271 

Convection  currents  are  more  easily  set  up  in  gases 
than  in  liquids.  They  are  utilized  in  heating  buildings 
by  a  hot-air  furnace.  Convection  currents  of  air  on  a 
large  scale  are  present  near  the  seacoast.  The  wind  is  a 
sea  breeze  during  the  day,  because  the  air  moves  in  from 
the  cooler  ocean  to  take  the  place  of  the  air  rising  over 
the  heated  land.  As  soon  as  the  sun  sets,  the  ground 
cools  rapidly  by  radiation,  and  the  air  over  it  is  cooler 
than  over  the  sea.  Hence  the  reversal  in  the  direction  of 
the  wind,  which  is  now  a  land  breeze. 

340.  Radiation.  —  When  one  stands  near  a  hot  stove,  one 
is  warmed  neither  by  heat  conducted  nor  conveyed  by  the 
air.     The  heat  energy  of  a  hot  body  is  constantly  passing 
into  space  as  radiant  energy  in  the  ether.     Radiant  energy 
becomes  heat  again  only  when  it  is  absorbed  by  bodies 
upon  which  it  falls.     Energy  transmitted  in  this  way  is, 
for  convenience,  referred  to  as  radiant  heat,  although  it  is 
transmitted  as  radiant  energy,  and  is  transformed  into 
heat   only  by  absorption.     Radiant   heat   and   light   are 
physically  identical,  but  are  perceived  through  different 
avenues  of  sensation.     Radiations  that  produce  sight  when 
received   though   the    e}^e   give   a   sensation   of    warmth 
through  the  nerves  of  touch,  or  heat  a  thermometer  when 
incident  upon  it.     The  long  ether  waves  do  not  affect  the 
eye,  but  they  heat  a  body  which  absorbs  them. 

341.  Laws  of  Heat  Radiation.  —  The  following  laws  have 
been  established  experimentally:  — 

I.  Radiation  proceeds  in  straight  lines.     This  law  is 
illustrated  in  the  use  of  fire   screens  and  sunshades,  and 
by  the  drop  in  temperature  when  the  sun  is  obscured  by 
dense  clouds. 

II.  The  amount  of  radiant  energy  received  by  a  body 


272  HEAT 

from  any  small  area  varies  inversely  as  the  square  of  Us 
distance  from  this  area  as  a  source.  This  law  is  the  same 
as  the  one  relating  to  the  intensity  of  illumination  in 
light. 

III.  Radiant  energy  is  reflected  from  a  polished  surface 
so  that  the  angles  of  incidence  and  reflection  are  equal. 
An  interesting  application  is  the  Ericsson  solar  engine. 
The  radiant  energy  of  the  sun  is  concentrated  by  large 
concave  reflectors  on  metal  pipes  filled  with  water.     The 
heat  is  great  enough  to  generate  steam  to  operate  a  small 
steam  engine. 

IV.  The  capacity  of  a  surf  ace  to  reflect  radiant  energy 
depends  both  on  the  polish  of  the  surface  and  the  nature 
of  the  material.    Polished  brass  is  one  of  the  best  reflectors, 
and  lampblack  is  the  poorest. 

V.  The  rate  at  which  the  temperature  of  a  cooling  body 
falls  by  radiation  is  proportional  to  the  excess  of  its  tem- 
perature over  that  of  the  surrounding  medium.     This  is 
known  as  Newton's  Law  of  Cooling ;    it   holds   approxi- 
mately for  small  differences  of  temperature  but  fails  when 
the  excess  is  large.     According  to  this  law  a  body  at  a 
temperature  of  30°  C.  cools  twice  as  fast  as  one  of  25°  C. 
in  air  at  20°  C.,  for  the  excess  10°,  in  the  first  case,  is  twice 
5°,  the  excess  in  the  second. 

VI.  The  transmission  of  radiant  heat  through  various 
substances  depends  on  the  wave  length  of  the  radiations, 
and  the  thickness  and  character  of  the  substance  itself. 
Those  transmitting  a  large  part  of  the  heat  energy,  as  rock 
salt,  are  said  to  be  diatliermanous :  those  absorbing  a  large 
part,  as  water  and  alum,  are  athermanous.     Glass  is  diather- 
manous  to  radiations  from  a  source  of  high  temperature, 


TRANSMISSION  OF  HEAT  273 

but  athermanous  to  radiations  from  sources  of  low  temper- 
ature. The  radiant  energy  from  the  sun  passes  readily 
through  the  atmosphere  of  the  earth,  warming  its  surface  ; 
but  the  radiations  from  the  earth  are  stopped  to  a  large 
extent  by  the  enveloping  atmosphere. 


Questions  and  Problems 

1.  Why  does  water  cool  faster  in  a  pan  than  in  a  pitcher  ? 

2.  Why  does  sawdust  keep  ice  from  melting  ? 

3.  Why  does  snow  protect  the  ground  from  freezing? 

4.  Why  does  the  "  thermos  "  bottle  keep  its  contents  hot  for  such 
a  long  time? 

5.  Why  is  glass  used  for  the  roofs  of  greenhouses  ? 

6.  What  principles  of  heat  are  applied  in  the  construction  of  a 
"  fireless  cooker  "  ? 

7.  Why  are  steam  pipes  in  the  basement  of  a  building  covered 
with  asbestos,  felt,  or  magnesia? 

8.  Should  the  surface  of  a  steam  or  water  radiator  be  polished  or 
rough? 

9.  Why  are  the  extremes  of  island  climates  less  than  elsewhere? 

10.  In  heating  water  why  should  the  heat  be  applied  at  the  bottom 
of  the  vessel  rather  than  at  the  top  ? 

11.  What  will  be  the  effect  on  the  reading  of  a  thermometer  to 
cover  the  bulb  with  a  wet  cloth  ? 

12.  In  what  way  does  a  grate  fire  heat  a  room? 

13.  A  thick  glass  tumbler  cracks  when  hot  water  is  poured  into  it. 
Why  does  not  a  thin  glass  beaker  crack  under  the  same  circumstances  ? 

14.  Why  are  clouds  a  protection  against  frost? 

15.  Why  is  the  boiling  point  of  water  in   the  boiler  of  a  steam 
engine  above  100°  C  ? 

16.  Why  will  a  moistened  finger  or  the  tongue  freeze  very  quickly 
to  a  piece  of  very  cold  iron,  but  not  to  a  piece  of  wood? 


274  HEAT 

VII.     HEAT  AND  WORK 

342.  Heat  from  Mechanical  Action.  —  strike  the  edge  of  a 
piece  of  flint  a  glancing  blow  with  a  piece  of  hardened  steel.     Sparks 
will  fly  at  each  blow. 

Pound  a  bar  of  lead  vigorously  with  a  hammer.  The  temperature 
of  the  bar  will  rise. 

In  the  cavity  at  the  end  of  a  piston  of  a  fire  syringe  place  a  small 
piece  of  tinder,  such  as  is  employed  in  cigar  lighters  (Fig.  286). 
Force  the  piston  quickly  into  the  barrel.  If  the  piston  is  immedi- 
ately withdrawn  the  tinder  will  probably  be  on  fire. 

These  experiments  show  that '  mechanical  energy  may 
be  transformed  into  heat.  Some  of  the  energy  of  the 
descending  flint,  the  hammer,  and  the  piston  has 
in  each  case  been  transferred  to  the  molecules  of 
the  bodies  themselves,  increasing  their  kinetic 
energy,  that  is,  raising  their  temperature. 

Savages  kindle  fire  by  rapidly  twirling  a  dry 
stick,  one  end  of  which  rests  in  a  notch  cut. in  a 
second  dry  piece.     The  axles  of  carriages  and  the 
bearings  in  machinery  are  heated  to  a  high  tem- 
perature   when    not    properly   lubricated.       The 
heating  of  drills  and  bits  in  boring,  the  heating 
of   saws  in   cutting   timber,  the   burning  of   the 
hands  by  a  rope  slipping  rapidly  through  them, 
the  stream  of  sparks  flying  from  an  emery  wheel, 
lg'         are  instances  of  the  same  kind  of  transformation; 
the  work   done  against  friction  produces  kinetic  energy 
in  the  form  of  heat. 

343.  The  Mechanical  Equivalent  of  Heat.  —  In  1840  Joule 
of  Manchester  in  England  began  a  series  of  experiments 
to  determine  the  numerical  relation  between  the  unit  of 
heat   and   the   foot   pound.     His   experiments    extended 
over  a  period  of  forty  years.     His  most  successful  method 


James  Watt  (1736-1819)  was  born  at  Greenock,  Scotland,  and 
was  educated  as  an  instrument  maker.     In  studying  the  defects  of 

the  steam  engines  then  in  use, 
he  was  led  to  make  many 
very  important  improvements, 
culminating  in  his  invention 
of  the  double-acting  steam 
engine.  He  invented  the  ball 
governor,  the  cylinder  jacket, 
the  D-valve,  the  jointed  paral- 
lelogram for  securing  recti- 
linear motion  to  the  piston, 
the  mercury  steam-gauge, 
and  the  water-gauge.  He  is 
also  to  be  credited  with  the  first 
compound  engine,  a  type  of  en- 
gine extensively  used  to-day. 


James  Prescott  Joule  (1818-1889),  the  son  of  a  Manchester 
brewer,  was  born  at  Salford, 
England.  He  became  known 
to  the  scientific  world  through 
his  contributions  in  heat,  elec- 
tricity, and  magnetism.  His 
greatest  achievement  was  es- 
tablishing the  modern  kinetic 
theory  of  heat  by  determining 
the  mechanical  equivalent  of 
heat.  His  experiments  on  this 
subject  were  continued 
through  a  period  of  forty  years. 
In  recognition  of  his  great  work 
he  was  presented  with  the 
Royal  Medal  of  the  Royal  So- 
ciety of  England  in  1852. 


HEAT  AND    WORK 


275 


consisted  in  measuring  the  heat  produced  when  a  meas- 
ured amount  of  work  was  expended  in  heating  water  by 
stirring  it  with  paddles  driven  by  weights  falling  through 
a  known  height.  His  final  result  was  that  772  ft.-lb. 
of  work,  when  converted  into  heat,  raise  the  temperature 
of  1  Ib.  of  water  1°  F.,  or  1390  ft.-lb.  for  1°  C.  The  later 
and  more  elaborate  researches  of  Rowland  in  1879  and  of 
Griffiths  in  1893  show  that  the  relation  is  778  ft.-lb.  for  1° 
F.,  or  427.5  kgm.-m.  for  1°  C. ;  that  is,  if  the  work  done  in 
lifting  427.5  kgm.  one  meter  high  is  all  converted  into  heat, 
it  will  raise  the  temperature  of  1  kgm.  of  water  1°  C.  This 
relation  is  known  as  the  mechanical  equivalent  of  heat.  Its 
value  expressed  in  absolute  units  is  4.19  x  107  ergs  per  calorie. 

344.  The  Steam  Engine.  —  The  most  important  devices 
for  the  conversion  of  heat  into  mechanical  work  are  the 
steam  engine  and  the  gas  engine.  The  former  in  its 
essential  features  was 
invented  by  James 
Watt.  In  the  recipro- 
cating steam  engine 
a  piston  is  moved  al- 
ternately in  opposite 
directions  by  the  pres- 
sure of  steam  applied 
first  to  one  of  its 
faces  and  then  to  the 
other.  This  recipro- 
cating or  to-and-fro 
motion  is  converted 
into  rotatory  motion 
by  the  device  of  a  connecting  rod,  a  crank,  and  a  flywheel. 

In  Fig.  287  are  shown  in  section  the  cylinder,  piston, 


276  HEAT 

and  slide  valve  of  a  simple  steam  engine.  The  piston  B 
is  moved  in  the  cylinder  A  by  the  pressure  of  the  steam  ad- 
mitted through  the  inlet  pipe  a.  The  slide  valve  d  works 
in  the  steam  chest  cc  and  admits  steam  alternately  to  the 
two  ends  of  the  cylinder  through  the  steam  ports  at  either 
end. 

When  the  valve  is  in  the  position  shown,  steam  passes 
into  the  right-hand  end  of  the  cylinder  and  drives  the 
piston  toward  the  left.  At  the  same  time  the  other  end 
is  connected  with  the  exhaust  pipe  ee  through  which  the 
steam  escapes,  either  into  the  air,  as  in  a  high-pressure 
non-condensing  engine,  or  into  a  large  condensing  chamber, 
as  in  a  low-pressure  condensing  engine. 

The  slide  valve  d  is  moved  by  the  rod  R,  connected  to 
an  eccentric,  which  is  a  round  disk  mounted  a  little  to  one 
side  of  its  center,  on  the  engine  shaft.  It  has  the  effect  of 
a  crank.  The  flywheel,  also  mounted  on  the  shaft  of  the 
engine,  has  a  heavy  rim  and  serves  as  a  store  of  energy 
to  carry  the  shaft  over  the  dead  points  when  the  piston  is 
at  either  end  of  the  cylinder.  There  is  in  the  flywheel  a 
give-and-take  of  energy  twice  every  revolution,  and  a  fairly 
steady  rotation  of  the  shaft  is  the  result. 

The  eccentric  is  set  in  such  a  way  that  the  rod  R  closes 
the  valve  admitting  steam  to  either  end  of  the  cylinder  be- 
fore the  piston  has  completed  its  stroke;  the  motion  of  the 
piston  is  continued  during  the  remainder  of  the  stroke  by 
the  expansive  force  of  the  steam. 

345.  The  Gas  Engine.  — The  gas  engine  is  a  type  of  in- 
ternal combustion  engine,  which  includes  motors  using  gas, 
gasoline,  kerosene,  or  alcohol  as  fuel.  The  fuel  is  intro- 
duced into  the  cylinder  of  the  engine,  either  as  a  gas  or  a 
vapor,  mixed  with  the  proper  quantity  of  air  to  produce  a 


HEAT  AND    WORK 


277 


good  explosive  mixture.  The  mixture  is  ignited  at  the 
right  instant  by  means  of  an  electric  spark.  The  explosion 
and  the  expansive  force  of  the  hot  gases  drive  the  piston 
forward  in  the  cylinder. 

In  the  four-cycle  type  of  gas  engine,  the  explosive  mix- 
ture is  drawn  in  and  ignited  in  each  cylinder  only  every 
other  revolution  of  the  engine,  while  in  the  two-cycle  type 
an  explosion  occurs  every  revolution.  The  former  type  is 
used  in  most  motor  car  engines,  and  the  latter  in  small 
motor  boats. 

The  operation  of  a  four-cycle  engine  is  illustrated  in  1, 
2,  3,  and  4  of  Fig.  288,  which  shows  the  four  steps  in  a 
complete  cycle.  The  inlet  valve 
a  and  the  exhaust  valve  b  are 
operated  by  the  cams  c  and  d. 
Both  valves  are  kept  normally 
closed  by  springs  surrounding 
the  valve  stems.  The  small 
shafts  to  which  the  two  cams 
are  fixed  are  driven  by  the  spur 
wheel  e  on  the  shaft  of  the 
engine.  This  wheel  engages 
with  the  two  larger  spur  wheels 
on  the  cam  shafts,  each  having 
twice  as  many  teeth  as  e  and 
forming  with  it  a  two-to-one 
gear,  so  that  c  and  d  rotate 
once  in  every  two  revolutions 
of  the  crank  shaft.  The  piston  m  has  packing  rings ;  h  is 
the  connecting  rod,  k  the  crank  shaft,  and  I  the  spark  plug. 

In  diagram  1  the  piston  is  descending  and  draws  in  the 
charge  through  the  open  valve  a ;  in  2  both  valves  are 
closed  and  the  piston  compresses  the  explosive  charge ; 


Fig.  288 


278 


HEAT 


about  the  time  the  piston  reaches  its  highest  point,  the 
charge  is  ignited  by  a  spark  at  the  spark  plug,  and  the 
working  stroke  then  takes  place,  as  in  3,  both  valves  re- 
maining closed;  in  4  the  exhaust  valve  b  is  opened  by  the 
cam  6?,  and  the  products  of  the  combustion  escape  through 
the  muffler,  or  directly  into  the  open  air.  The  piston  has 
now  traversed  the  cylinder  four  times,  twice  in  each  di- 
rection, and  the  series  of  operations  begins  again. 

Fig.  289  is  a  section  of  a  two-cycle  engine.  During 
the  up-stroke  of  the  piston  P  a  charge  is  drawn  through 
A  into  the  crank  case  O.  At  the  same  time  a  charge  in 
the  cylinder  is  compressed  and  is 
ignited  by  a  spark  when  the  com- 
pression is  greatest.  The  piston  is 
forced  down,  and  when  it  passes 
the  port  E  the  exhaust  takes  place. 
When  the  admit  port  I  is  passed, 
a  charge  enters  from  the  crank 
case.  To  prevent  this  charge  from 
passing  across  and  escaping  at  E,  it 
is  made  to  strike  against  a  projec- 
tion B  on  the  piston,  which  de- 
flects it  upward.  The  momentum  of 
the  balance  wheel  carries  the  piston 
upward,  compresses  the  charge,  and 


Fig.  289 


draws  a  fresh  charge  into  the  crank  case.  The  piston  has 
now  traversed  the  cylinder  twice,  once  in  each  direction, 
and  the  same  series  of  operations  is  again  repeated. 

346.  The  Aeroplane.  —  If  a  large  flat  surface,  placed 
obliquely  to  the  ground,  be  moved  along  somewhat  rapidly, 
it  will  be  lifted  upward  by  the  vertical  component  of  the 
reaction  of  the  air  against  it,  just  as  a  kite  is  lifted  (§  108). 


QUESTIONS  AND  PROBLEMS 


279 


This  is  the  principle  applied  in  the  aeroplane.  In  this 
device,  whether  monoplane  or  biplane,  large  bent  surfaces 
attached  to  a  stout  light  frame  are  driven  through  the  air 
by  rapidly  rotating  propellers  operated  by  a  powerful  gas- 
oline engine,  just  as  a  steamboat  is  driven  through  the 
water.  By  means  of  suitable  levers  under  control  of  the 
driver,  these  planes,  or  certain  auxiliary  planes,  can  be  set 
at  an  angle  to  the  stream  of  air  against  which  they  are 
propelled.  Then,  as  in  the  kite,  they  rise  through  the  air 


Fig.  290 

by  the  action  of  the  vertical  component  of  the  force  of  the 
air  against  their  under  surfaces.  Vertical  planes  are 
attached  to  the  same  frame  to  serve  as  rudders  in  steering 
either  to  the  right  or  the  left;  movements  either  up  or 
down  are  regulated  by  the  inclination  of  the  auxiliary  or 
elevating  planes  as  already  stated.  Fig.  290  illustrates 
one  of  the  many  types  of  aeroplanes  now  in  use. 

Questions  and  Problems 

1.   Why  does  the  temperature  of  the  air  under  the  bell  jar  of  an 
air  pump  fall  when  the  pump  is  worked? 


280  HEAT 

2.  Is  there  a  difference  in  the  temperature  of  the  steam  as  it 
enters  a  steam  engine  and  as  it  leaves  at  the  exhaust  ?     Explain. 

3.  Lead  bullets  are  sometimes  melted  when  they  strike  a  target. 
Explain. 

4.  Does  warm  clothing  keep  the  cold  out ?    What  does  it  do? 

5.  Describe  the  movements  of  the  air  in  a  room  heated  by  a  stove. 

6.  Is  there  any  less  moisture  in  the  air  after  it  has  passed  through 
a  heated  furnace  into  a  room  than  there  was  before? 

7.  A  mass  of  200  gm.  moving  with  a  velocity  of  50  m.  per  second, 
is  suddenly  stopped.     If  all  its  energy  is  converted  into  heat,  how 
many  calories  would  it  be  ? 

(Kinetic  energy  =  |  mv2,  and  a  calorie  =  4.19  x  107  ergs.) 

8.  If  all  the  potential  energy  of  a  300  kgm.  mass  of  rock  at  an 
elevation  of  250  m.  is  converted  into  heat  by  falling,  how  many  cal- 
ories would  be  produced  ? 

9.  How  high  could  a  200  gm.  weight  be  lifted  by  the  heat  required 
to  melt  the  same  mass  of  ice,  if  all  the  heat  could  be  utilized  for  the 
purpose  ? 

10.  If  the  average  pressure  of  the  steam  in  the  cylinder  of  an 
engine  is  100  Ib.  per  sq.  in.,  and  the  area  of  the  piston  is  80  sq.  in., 
the  stroke  one  foot,  and  if  the  engine  makes  two  revolutions  per 
second,  how  many  horse  powers  would  it  develop  ? 


CHAPTER   X 


MAGNETISM 
I.  MAGNETS  AND  MAGNETIC  ACTION 

347.  Natural  Magnets.  —  Black  oxide  of  iron,  commonly 
called  magnetite,  is  widely  distributed  and  is  sometimes 
found  to  possess  the  property  of  attracting  small  pieces  of 
iron.     At  a  very  early  date  such  pieces  of  iron  ore  were 
found  near  Magnesia  in  Asia  Minor,  and  they  were  there- 
fore called  magnet1^  stones  and  later  magnets.     They  are 
now  known  as  natural  magnets,  and  the  properties  peculiar 
to  them  as  magnetic  properties. 

Dip   a   piece   of   natural   magnet  into  iron  filings;  they  will  ad- 
here to   it    in  tufts,    not  uniformly   over  its   surface,  but  chiefly  at 
the  ends  and  on  projecting  edges  (Fig.  291). 
Suspend   a  piece  of   natural   magnet  by  a 
piece  of  untwisted  thread  (Fig.  292).     Note  its 

position,  then  disturb 

it   slightly,  and  again 

let  it'  come  to  rest.     It 

will  be  found   that  it 

invariably   returns   to 

the  same  position,  the 
line  connecting  the  two   ends   to  which  the  filings  chiefly  adhered 
in  the  preceding  experiment  lying  north  and  south. 

This  directional  property  of  the  natural  magnet  was 
early  turned  to  account  in  navigation,  and  secured,  for  it 
the  name  of  lodestone  (leading-stone). 

348.  Artificial  Magnets.  —  Stroke  the  blade  of  a  pocket  knife 
from  end  to  end,  and  always  in  the  same  direction,  with  one  end  of  a 

281 


Fig.  291 


Fig.  292 


282  MAGNETISM 

lodestone.     Touch  it  to  iron  filings ;  they  will  cling  to  its  point  as 
they  did  to  the  lodestone.     The  knife  blade  has  become  a  magnet. 

Use  the  knife  blade  of  the  last  experiment  to  stroke  another  blade. 
This  second  blade  will  also  acquire  magnetic  properties,  and  the  first 
one  has  suffered  no  loss. 

Artificial  magnets,  or  simply  magnets,  are  bars  of  hard- 
ened steel  that  have  been  made  magnetic  by  the  applica- 
tion of  some  other  magnet  or 
magnetizing  force.  The  form 
of  artificial  magnets  most  com- 
monly met  with  are  the  bar  and 
the  horseshoe  (Fig.  293). 


349.    Magnetic    Substances.  — 

Any  substance  that  is  attracted 

by  a  magnet  or  that  can  be  magnetized  is  a  magnetic 
substance.  Faraday  showed  that  most  substances  are  in- 
fluenced by  magnetism,  but  not  all  in  the  same  way  nor 
to  the  same  degree.  Iron,  nickel,  cobalt,  and  manganese 
are  powerfully  attracted  by  magnets  and  are  said  to  be 
magnetic;  bismuth,  antimony,  and  zinc  act  as  if  they  are 
repelled  by  magnets  and  they  are  called  diamagnetic. 

350.    Polarity.  —  Roll  a  bar  magnet  in  iron  filings.     It  will  be- 
come   thickly   covered    with    the    filings   near    its   ends.     Few,   if 
any,   will    adhere   at    the  middle 
(Fig.  294). 

The  experiment  shows  that 
the  greater  part  of  the  mag-  Fi     294 

netic    attraction    is    concen- 
trated at  or  near  the  ends  of  the  magnet.     They  are  called 
its  poles,  and  the  magnet  is  said  to  have  polarity.    The  line 
joining  the  poles  of  a  long  slender   magnet   is  its  mag- 
netic axis. 


Michael  Faraday,  1791-1867,  was  born  near  London,  England. 
He  was  the  son  of  a  blacksmith  and  received  but  little  schooling, 
being  apprenticed  to  a  bookbinder  when  only  thirteen  years  of 
age.  While  employed  in  the  bindery  he  became  interested  in 
reading  such  scientific  books  as  he  -found  there.  Later  he  applied 
to  Sir  Humphry  Davy  for  consideration  and  was  made  Davy's 
assistant.  From  this  time  his  rise  was  rapid  ;  in  1816  he  published 
his  first  scientific  memoir;  in  1824  he  became  a  member  of  the 
Royal  Society;  in  1825  he  was  elected  director  of  the  Royal 
Institution  ;  '  in  1831  he  announced  the  discovery  of  magneto- 
electric  induction,  the  most  important  scientific  discovery  of  any 
age,  In  1833  he  was  elected  professor  of  chemistry  in  the  Royal 
Institution.  He  was  a  remarkable  experimenter  and  a  most  inter- 
esting lecturer,  and  amid  all  his  wonderful  achievements,  he  was 
utterly  wanting  in  vanity. 


MAGNETS  AND  MAGNETIC  ACTION  283 

351.  North  and   South   Poles.  —  Straighten  a  piece  of  watch 
spring  8  or  10  cm.  long,  stroke  it  from  end  to  end  with  a  magnet,  and 
float   it   on  cork  in  a 

vessel  of  water  (Fig. 
295).  It  will  turn  from 
any  other  position  to  a 
north  and  south  one, 
and  invariably  with 
the  same  end  north.  Fig.  295 

The  end  of  a  magnet  pointing  toward  the  north  is  called 
the  north-seeking  pole,  and  the  other  the  south-seeking  pole. 
They  are  commonly  called  simply  the  north  pole  and  the 
south  pole. 

352.  Magnetic  Needle.  —  A  slender  magnetized  bar,  sus- 
pended by  an  untwisted  fiber  or  pivoted  on  a  point  so  as  to 
have  freedom  of  motion  about  a  vertical  axis  is  a  magnetic 

needle  (Fig.  296).  The  di- 
rection in  which  it  comes  to 
rest  without  torsion  or  friction 
is  called  the  magnetic  meridian. 

Fasten  a  fiber  of  unspun  silk  to 
a  piece  of  magnetized  watch  spring 
about  2  cm.  long  so  that  it  will  hang 
horizontally.  Suspend  it  inside  a 
wide-mouthed  bottle  py  attaching 
the  fiber  to  a  cork  fitting  the  mouth 
of  the  bottle.  The  little  magnetic 
needle  will  then  be  protected  from 

currents  of  air.     It  may  be  made  visible  at  a  distance  by  sticking 

fast  to  it  a  piece  of  thin  white  paper. 

353.  Magnetic  Transparency.  —  Cover  the  pole  of  a  strong  bar 
magnet  with  a  thin  plate  of  glass.     Bring  the  face  of  the  plate  oppos- 
ite the  pole  in  contact  with  a  pile  of  iron  tacks.     A  number  will  be 
found   to  adhere,  showing  that  the  attraction  takes  place  through 
glass.     In  like  manner,  try  thin  plates  of  mica,  wood,  paper,  zinc, 


284  MAGNETISM 

copper,  and  iron.     No  perceptible  difference  will  be  seen  except  in 
the  case  of  the  iron,  where  the  number  of  tacks  lifted  will  be  much  less. 

Magnetic  force  acts  freely  through  all  substances  except 
those  classified  as  magnetic.  Soft  iron  serves  as  a  more  or 
less  perfect  screen  to  magnetism.  Watches  may  be  pro- 
tected from  magnetic  force  that  is  not  too  strong  by  means 
of  an  inside  case  of  soft  sheet  iron. 

354.  First  Law  of  Magnetic  Action.  —  Magnetize  a  piece  of  large 
knitting-needle,  about  four  inches  long,  by  stroking  it  from  the  middle 
to  one  end  with  the  north  pole  of  a  bar  magnet,  and  then  from  the 
middle  to  the  other  end  with  the  south  pole.     Repeat  the  operation 
several  times.     Present  the  north  pole  of  the  magnetized  knitting- 
needle  to  the  north  pole  of  the  needle  suspended  in  the  bottle.     The 
latter  will  be  repelled.     Present  the  same  pole  to  the  south  pole  of  the 
little  magnetic  needle ;  it  will  be  attracted.     Repeat  with  the  south 
pole  of  the  knitting-needle  and  note  the  deflections. 

The  results  may  be  expressed  by  the  following  law  of 
magnetic  attraction  and  repulsion:  - 

Like  magnetic  poles  repel  and  unlike  magnetic  poles 
attract  each  other. 

355.  Testing  for  Polarity. — The  magnetic  needle  affords 
a  ready  means  of  ascertaining  which  pole  of  a  magnet  is 

the  north  pole,  for  the  north  pole  of 
the  magnet  is  the  one  that  repels  the 
north  pole  of  the  magnetic  needle. 
Repulsion  is  the  only  sure  test  of  po- 
larity for  reasons  that  will  appear  in 
the  experiments  that  follow. 

356.   Induced  Magnetism.  —  Hold  verti- 

P.     297  T|       cally  a  strong  bar  magnet  and  bring  up  against 

its  lower  end  a  short  cylinder  of  soft  iron.  It 
will  adhere.  To  the  lower  end  of  this  one  attach  another,  and  so  on  in  a 
series  of  as  many  as  will  stick  (Fig.  297).  Carefully  detach  the  magnet 
from  the  first  piece  of  iron  and  withdraw  it  slowly.  The  pieces  of 
iron  will  all  fall  apart. 


MAGNETS  AND  MAGNETIC  ACTION  285 

The  small  bars  of  iron  hold  together  because  they  become 
temporary  magnets.  Magnetism  produced  in  magnetic 
substances  by  the  influence  of  a  magnet  near  by  or  in 
contact  with  them  is  said  to  be  induced,  and  the  action  is 
called  magnetic  induction.  Magnetic  induction  precedes 
attraction. 


357.    Unlike  Polarity  Induced.  —  Place  a  bar  magnet  in  line 
with  the  magnetic  axis  of  a  magnetic  needle,  with  its  north  pole  as 
near   as  possible   to  the  north 
pole  of  the  needle  without  ap-      ^^^^^^__ 
preciably  repelling  it  (Fig.  298).  "IT? 

Insert  a  bar  of  soft  iron  between 
the  magnet  and  the  needle. 
The  north  pole  of  the  needle 
will  be  immediately  repelled.  Fig.  298 


^^ja^lp:. 


The  repulsion  of  the  north  pole  of  the  needle  by  the  end 
of  the  soft  iron  bar  next  to  it  shows  that  this  end  of  the 
bar  has  acquired  a  polarity  the  same  as  that  of  the  magnet, 
that  is,  north  polarity.  Then  the  other  end  adjacent  to 
the  magnet  must  have  acquired  the  opposite  polarity. 

When  a  magnet  is  brought  near  a  piece  of  iron,  the  iron 
is  magnetized  by  induction,  and  there  is  attraction  because 
the  adjacent  poles  are  unlike.  When  a  bunch  of  iron 
filings  or  tacks  adhere  to  a  magnet,  each  filing  or  tack 
becomes  a  magnet  and  acts  inductively  on  the  others 
and  all  become  magnets.  Weak  magnets  may  have  their 
polarity  reversed  by  the  inductive  action  of  a.  strong 
magnet. 

358.  Permanent  and  Temporary  Magnetism.  —  When  a 
piece  of  hardened  steel  is  brought  near  a  magnet,  it 
acquires  magnetism  as  the  piece  of  soft  iron  does  under 
the  same  conditions;  but  the  steel  retains  its  magnetism 
when  the  magnetizing  force  is  withdrawn,  while  the  soft 


286  MAGNETISM 

iron  does  not.  In  the  experiment  of  §  356  the  soft  iron 
ceases  to  be  a  magnet  when  removed  to  a  distance  from 
the  bar  magnet.  In  addition,  therefore,  to  the  permanent 
magnetism  exhibited  by  the  magnetized  steel,  we  have 
temporary  magnetism  induced  in  a  bar  of  soft  iron  when  it 
is  brought  near  a  magnet  or  in  contact  with  it. 

II.   NATURE  OF  MAGNETISM 

359.  Magnetism   a  Molecular  Phenomenon.  —  If  a  piece  of 
watch  spring  be  magnetized  and  then  heated  red  hot,  it  will  lose  its 
magnetism  completely. 

A  magnetized  knitting-needle  will  not  pick  up  as  many  tacks  after 
being  vibrated  against  the  edge  of  a  table  as  it  did  before. 

A  piece  of  moderately  heavy  and  very  soft  iron  wire  of  the  form 
shown  in  Fig.  299  can  be  magnetized  by  stroking  it  gently  with  a  bar 

magnet.     If   given   a  sudden   twist,   it 
loses  at   once    all   the    magnetism   im- 
Fig.  299  parted  to  it. 

A  piece  of  watch  spring  attracts  iron 

filings  only  at  its  ends.  If  broken  in  two  in  the  middle,  each  half 
will  be  a  magnet  and  will  attract  filings,  two  new  poles  having  been 
formed  where  the  original  magnet  was  neutral.  If  these  pieces  in 
turn  be  broken,  their  parts  will  be  magnets.  If  this  division  into 
separate  magnets  be  conceived  to  be  carried  as  far  as  the  molecules, 
they  too  would  probably  be  magnets. 

It  is  worthy  of  notice  that  magnetization  is  facilitated 
by  jarring  the  steel,  or  by  heating  it  and  letting  it  cool 
under  the  influence  of  a  magnetizing  force.  If  an  iron 
bar  is  rapidly  magnetized  and  demagnetized,  its  tempera- 
ture is  raised.  A  steel  rod  is  slightly  lengthened  by  mag- 
netization and  a  faint  click  may  be  heard  if  the  magnetiza- 
tion is  sudden. 

360.  Theory  of  Magnetism.  —  The  facts  of  the  preceding 
article  indicate  that  the  seat  of  magnetism  is  the  molecule, 
that  the  individual  molecules  are  magnets,  that  in  an  un- 


THE  MAGNETIC  FIELD 


287 


magnetized  piece  of  iron  the  poles  of  the  molecular  magnets 
are  turned  in  various  directions,  so  that  they  form  stable 
combinations  or  closed  magnetic  chains,  and  hence  exhibit 
no  magnetism  external  to  the  bar.  In  a  magnetized  bar 
the  larger  portion  of  the  molecules  have  their  magnetic 
axes  pointing  in  the  same  direction,  the  completeness  of 
the  magnetization  depending  on  the  completeness  with 
which  this  uniformity  of  direction  is  secured. 

III.    THE  MAGNETIC  FIELD 

361.  Lines  of  Magnetic  Force.  —  Place  a  sheet  of  paper  over  a 
small  bar  magnet  and  sift  iron  filings  evenly  over  it  from  a  bottle 
with  a  piece  of  gauze  tied  over  the  mouth,  tapping  the  paper  gently 
to  aid  the  filings  in  arranging  themselves  under  the  influence  of  the 
magnet.     They  will  cling  together  in  curved  lines,  which  diverge 
from  one  pole  of  the  magnet  and  meet  again  at  the  opposite  pole. 

These  lines  are  called  lines  of  magnetic  force  or  of  mag- 
netic induction.  Each  particle  of  iron  becomes  a  magnet 
by  induction  ;  hence  the  lines  of  force  are  the  lines  along 
which  magnetic  induction  takes  place. 

362.  Magnetic  Fields.  —  A  magnetic  field  is  the  space 
around  a  magnet  in  which  there  are  lines  of  magnetic 


Fig.  300 

force.     Fig.  300  was  made  from  a  photograph  of  the  magnetic  field 
of  a  bar  magnet  in  a  plane  passing  through  the  magnetic  axis.    These 


288 


MAGNETISM 


lines  of  force  branch  out  from  the  north  pole,  curve  round  through  the  air 

to  the  south  pole,  and  complete  their  circuit  through  the  magnet  itself. 

Fig.  301  shows  the  field  about  two  bar  magnets  placed  with  their 

unlike  poles  adjacent  to 
each  other.  Many  of  the 
lines  from  the  north  pole 
of  the  one  extend  across 
to  the  south  pole  of  the 
other,  and  this  connec- 
tion denotes  attraction. 

Fig.  302  shows  the  field 
about  two  bar  magnets 
with  their  like  poles  ad- 
jacent to  each  other. 
None  of  the  lines  spring- 
ing from  either  pole  ex- 


Fig.  301 


tend  across  to  the  neighboring  pole  of  the  other  magnet.     This  is 
a  picture  of  magnetic  repulsion. 

363.  Properties  of  Lines  of  Force.  —  Lines  of  magnetic 
force  have  the  following  properties :   (#)  They  are  under 
tension,  exerting  a  pull  in  the  direction  of  their  length ; 
(5)  they  spread  out  as  if  repelled  from  one  another  at 
right  angles  to  their  length ;    (c)  they  never  cross  one 
another;   (tT)  they  form  continuous  closed  curves. 

364.  Direction  of  Lines  of  Force.  —  Hold  a  mounted  magnetic 
needle  about  1  cm.  long  near  a  bar  magnet.     It  will  place  itself  tan- 
gent to  the  line  of  force 

passing  through  it. 

Suspend  by  a  fine 
thread  about  60  cm.  long 
a  strongly  magnetized 
sewing  needle  with  its 
north  pole  downward. 
Bring  this  pole  of  the 
needle  over  the  north 
pole  of  a  horizontal  bar 
magnet  (Fig.  303).  It 
will  be  repelled  and  will 


THE  MAGNETIC  FIELD 


289 


move   along  a   curved  line  of    force    toward  the  south  pole  of  the 
magnet. 

The  direction  of  a  line  of  force  at  any  point  is  that  of 
a  line  drawn  tangent  to  the  curve  at  that  point,  and  the 
positive  direction  is 
that  in  which  a  north 
pole  is  urged.  Since 
the  north  pole  of  a 
magnetic  needle  is  re- 
pelled by  the  north 
pole  of  a  bar  magnet, 
an  observer  stand- 
ing with  his  back  to 

the     north     pole     of 

i      T        •  Fig.  303 

a    magnet    looks    in 

the  direction  of  the  lines  of  force  coming  from  that  pole. 

365.    Permeability.  — Place  a  piece  of  soft  iron  near  the  pole  of 
a  bar  magnet  and  map  out  the  field  with  iron  filings.     The  lines  are 

displaced  by  the 
iron  and  are  gath- 
ered into  it  (Fig. 
304). 

When  iron  is 
placed  in  a  mag- 
netic field,  the 
lines  of  force  are 
concentrated  by 
it.  This  prop- 
erty possessed 
by  iron,  when 
placed  in  a  magnetic  field,  of  concentrating  the  lines 
of  force  and  increasing  their  number,  is  known  as  per- 
meability. The  superior  permeability  of  soft  iron  ex- 


Fig.  304 


290  MAGNETISM 

plains  the  action  of  magnetic  screens  (§  353).  In  the 
case  of  the  watch  shield,  the  lines  of  force  follow  the  iron 
and  do  not  cross  it ;  the  watch  is  thus  protected  from  mag- 
netism because  the  lines  of  force  do  not  pass  through  it. 

IV.    TERRESTRIAL  MAGNETISM 

366.  The  Earth  a  Magnet.  —  Support  a  thoroughly  annealed 
iron  rod  horizontally  in  an  east-and-west  line  and  test  it  for  polarity. 
It  should  show  no  magnetism.     Now  place  it  north  and  south  with 
the  north  end  about  70°  below  the  horizontal.     While  in  this  posi- 
tion, tap  it  with  a  hammer  and  then  test  it  for  polarity.     The  lower 
end  will  be  found  to  be  a  north  pole  and  the  upper  end  a  south  pole. 
Turn  the  rod  end  for  end,  hold  in  the  former  position,  and  tap  again 
with  a  hammer.     The  lower  end  will  again  become  a  north  pole  and 
the  magnetism  has  been  reversed. 

This  experiment  shows  that  the  earth  acts  as  a  magnet 
on  the  iron  rod  and  magnetizes  it  by  induction.  Similarly, 
iron  objects,  such  as  a  stove,  a  radiator,  vertical  steam 
pipes,  iron  columns  and  hitching  posts  become  magnets 
with  the  lower  end  a  north  pole.  The  inductive  action  of 
the  earth  as  a  magnet  accounts  for  the  magnetism  of  nat- 
ural magnets. 

367.  Magnetic     Dip.  —  Thrust    two     unmagnetized    knitting 
needles  through   a  cork   at  right  angles  to  each  other  (Fig.  305). 

Support  the  apparatus  on  the  edges  of 
two  glasses,  with  the  axis  in  an  east-and- 
west  line,  and  the  needle  adjusted  so  as 
to  rest  horizontally.  Now  magnetize  the 
needle,  being  careful  not  to  displace  the 
cork.  It  will  no  longer  assume  a  hori- 
zontal position,  the  north  pole  dipping 
down  as  if  it  had  become  heavier. 

The    inclination    or    dip    of    a 

needle  is  the  angle  its  magnetic  axis  makes  with  a 
horizontal  plane.  A  needle  mounted  so  as  to  turn  about 


TERRESTRIAL   MAGNETISM 


291 


a  horizontal  axis  through  its  center  of  gravity  is  a  dip- 
ping needle  (Fig.  306).  The  dip  of  the  needle  at  the 
magnetic  poles  of  the  earth  is  90°,  at  the  magnetic 
equator,  0°.  In  1907  Amundsen  placed  the  magnetic 
pole  of  the  northern  hemisphere  in 
latitude  75°  5'  N.  and  longitude  96°  47' 
W.  The  magnetic  pole  of  the  south- 
ern hemisphere  is  probably  near  lati- 
tude 73°  S.  and  longitude  150°  E. 

.Isoclinic  lines  are  lines  on  the 
earth's  surface  passing  through 
points  of  equal  dip.  They  are  irreg- 
ular in  direction,  though  resembling 
somewhat  parallels  of  latitude. 

368.    Magnetic    Declination.  —  The 

magnetic  poles  of  the  earth   do   not  Fi&'  306 

coincide  with  the  geographical  poles,  and  consequently 
the  direction  of  the  magnetic  needle  is  not  in  general 
that  of  the  geographical  meridian.  The  angle  between 
the  direction  of  the  needle  and  the  meridian  at  any 
place  is  the  magnetic  declination.  To  Columbus  be- 
longs the  undisputed  discovery  that  the  declination  is 
different  at  different  points  on  the  earth's  surface.  In 
1492  he  discovered  a  place  of  no  decimation  in  the  At- 
lantic Ocean  north  of  the  Azores.  The  declination  at 
any  place  is  not  constant,  but  changes  as  if  the  magnetic 
poles  oscillate,  while  the  mean  position  about  which  they 
oscillate  is  subject  to  a  slow  change  of  long  period.  The 
annual  change  on  the  Pacific  coast  is  about  4',  and  in 
New  England  about  3;.  At  London  in  1660  the  magnetic 
declination  was  zero,  and  it  attained  its  maximum  westerly 
value  of  24°  in  1810  ;  in  1900  it  had  decreased  again  to  15°. 


292  MAGNETISM 

369.  Agonic  Lines.  —  Lines  drawn  through  places  where 
the  needle  points  true  north  are  called  agonic  lines.  The 
one  in  North  America  runs  from  the  magnetic  pole  south- 
ward across  the  eastern  end  of  Lake  Superior,  thence  near 
Lansing,  Michigan,  Columbus,  Ohio,  through  West  Vir- 
ginia and  South  Carolina,  and  it  leaves  the  mainland  near 
Charleston  on  its  way  to  the  magnetic  pole  in  the  south- 
ern hemisphere.  East  of  this  line  the  needle  points  west 
of  north;  west  of  it,  it  points  east  of  north.  Lines  pas- 
sing through  places  of  the  same  declination  are  called 
isogonic  lines. 

Questions 

1.  How  would  you  determine  which  is  the  north  pole  of  a  magnet? 

2.  Given  two  steel  bars  exactly  alike  in  every  respect  except  that 
one  is  magnetized.     Select  the  magnetized  one. 

3.  How  would  you  magnetize  a  sewing  needle  so  that  the  eye  end 
shall  be  the  north  pole? 

4.  Given  two  magnets  that  look  alike  in  every  respect.     Deter- 
mine which  one  is  the  more  strongly  magnetized. 

5.  Give  two  reasons  for  calling  the  earth  a  great  magnet. 

6.  If  a  magnet  attracts  iron,  why  does  not  a  floating  iron  vessel  if 
left  to  itself  move  toward  the  north? 

7.  Does  it  make  any  difference  in  the  construction  of  a  pocket 
compass  whether  it  is  mounted  in  a  brass  case  or  an  iron  one  ? 

8.  Will  the  steel  I-beams  supporting  a  floor  have  any  effect  on  the 
indications  of  a  magnetic  needle  near  them? 

9.  If  the  earth's  magnetism  tends  to  make  the  north  pole  of  a 
magnetic  needle  dip  downward,  how  is  it  that  compass  needles  are 
horizontal? 

10.  In  what  direction  does  the  magnetic  needle  point  at  the  mag- 
netic pole  in  the  northern  hemisphere?  In  what  direction  does  a  dip- 
ping needle  stand?  Is  the  polarity  of  the  earth's  magnetism  in  the 
northern  hemisphere  the  same  as  that  of  the  north  pole  of  a  magnet 
or  that  of  the  south  pole  ? 


CHAPTER   XI 


ELECTROSTATICS 
I.    ELECTRIFICATION 

370.  Electrical  Attraction.  —  Rub  a  dry  flint  glass  rod  with  a  silk 
pad  and  bring  it  near  a  pile  of  pith  balls  or  bits  of  paper.  They  will 
at  first  be  attracted  and  then  repelled  (Fig.  307). 

The  simple  fact  that  a  piece  of  amber  (a  fossil  gum) 
rubbed  with  a  flannel  cloth,  acquires  the  property  of  at- 
tracting bits  of  paper, 
pith,  or  other  light 
bodies,  has  been  known 
since  about  600  B.C.; 
but  it  seems  not  to  have 
been  known  for  the 
following  2200  years 
that  any  bodies  except 
amber  and  jet  were 
capable  of  this  kind  of 
excitation.  About  1600  Gilbert  discovered  that  a  large 
number  of  substances  possess  the  same  property.  These 
he  called  electrics  (from  the  Greek  word  for  amber,  elec- 
tron). A  body  excited  in  this  manner  is  said  to  be 
electrified,  its  condition  is  one  of  electrification,  and  the 
invisible  agent  to  which  the  phenomenon  is  referred  is 
electricity. 

371.  Electrical  Repulsion.  —  Suspend  several  pith  balls  from  a 
glass  rod  or  insulated  hook  (Fig.  308).  Touch  them  with  an  electrified 


Fig.  307 


294 


ELECT R  OSTA  TICS 


glass  tube.     They  are  at  first  attracted  but  they  soon  fly  away  from  the 
tube  and  from  one  another.     When  the  tube  is  removed  to  a  distance, 

the  balls  no  longer  hang  side  by  side, 
but  keep  apart  for  some  little  time.  If 
we  bring  the  hand  near  the  balls  they 
will  move  toward  it  as  if  attracted, 

showing  that  the  balls  are  electrified. 

« 

This  experiment  shows  that 
bodies  become  electrified  by  com- 
ing in  contact  with  electrified 
bodies,  and  that  electrification 
may  show  itself  by  repulsion  as 
Fig.  308  well  as  by  attraction. 


372.  Attraction  Mutual.  —  Electrify  a  flint  glass  tube  by  fric- 
tion with  silk,  and  hold  it  near  the  end  of  a  long  wooden  rod  rest- 
ing in  a  wire  stirrup  suspended  by  a  silk  thread  (Fig.  309).     The 
suspended  rod  is  attracted.    Now,  replace  the  rod  by  the  electrified 
tube.     When  the  rod   is  held  near   the   rubbed  end  of   the   glass 
tube,  the  latter  moves  as  if  attracted  by 

the  former. 

The  experiment  teaches  that  each 
body  attracts  the  other;  that  is, 
that  the  action  is  mutual. 

373.  Two  Kinds  of  Electrification.— 

Rub  a  glass  tube  with  silk  and  suspend  it 
as  in  Fig.  309.  Rub  a  second  glass  tube 
and  hold  it  near  one  end  of  the  suspended 
one.  The  suspended  tube  will  be  re- 
pelled. Bring  near  the  suspended  tube  a  rod  of  sealing  wax  rubbed 
with  flannel.  The  suspended  tube  is  now  attracted.  Repeat  these 
tests  with  an  electrified  rod  of  sealing  wax  in  the  stirrup  instead  of 
the  glass  tube.  The  electrified  sealing  wax  will  repel  the  electrified 
sealing  wax,  but  there  will  be  attraction  between  the  sealing  wax  and 
the  glass  tube. 


Fig.  309 


ELECTRIFICA  TION 


295 


The  experiment  shows  that  there  are  two  kinds  of  electri- 
fication: one  developed  by  rubbing  glass  with  silk,  and 
the  other  by  rubbing  sealing  wax  with  flannel.  To  the 
former  Benjamin  Franklin  gave  the  name  positive  electrifi- 
cation; to  the  latter,  negative  electrification. 

374.  First  Law  of  Electrostatic  Action.  —  It  was  seen  in 
the  last  experiment  that  there  is  repulsion  between  electri- 
fied glass  tubes,  and  that  electrified  sealing  wax  attracts 
electrified  glass.     These  facts  are  expressed  by  the  follow- 
ing law :  — 

Like  electrical  charges  repel  each  other:  unlike  elec- 
trical charges  attract  each  other. 

375.  The   Electroscope.  —  An   instrument  for  detecting 
electrification  and  for  determining  its  kind  is  called  an 
electroscope.    Fig.  310 

illustrates  a  conven-' 
ient  form.  The  rec- 
tangular vessel  has 
two  opposite  faces  of 
glass,  metal  ends,  and 
a  wooden  or  ebonite 
base.  A  brass  rod, 
terminating  on  the 
outside  in  a  ball  or 
disk,  passes  through 
a  sulphur,  amber,  or 
shellac  plug  set  in  the 
top  of  the  case.  The 
indicating  system  Fig.  310 

consists    of    a    rigid 

piece  of  flat  brass,  to  which  is  attached  a  narrow  strip  of 
gold  leaf  or  other  metal  foil.     A  scale  drawn  on  mica 


296  ELECTROSTATICS 

is  sometimes  placed  adjacent  to  the  foil  to  measure  its 
displacement.  If  the  knob  be  touched  with  an  excited 
glass  tube,  the  foil  will  be  repelled  by  the  stem  because 
the  two  are  similarly  electrified. 

376.  Charging  an  Electroscope.  —  To  charge  an  electro- 
scope an  instrument  called  a  proof  plane  is  needed.     It 
consists   of   a   small   metal  disk  attached  to   an   ebonite 

handle  (Fig.  311).  To  use  it 
touch  the  disk  to  the  electrified 
body  and  then  apply  it  to  the 
knob  of  the  electroscope.  The 

angular  separation  of  the  foil  from  the  stem  will  indi- 
cate the  intensity  of  the  electric  charge  imparted.  This 
method  is  known  as  charging  by  contact  in  distinction  from 
charging  by  induction  to  be  described  later  (§  383). 

i 

377.  Both  Electrifications  Developed  Together.  —  Glue  a  small 

piece  of  fur  to  a  short  strip  of  wood.  Rub  a  rod  of  hard  rubber  or  a 
stick  of  sealing  wax  with  this  fur;  keep  the  two  in  contact  and  bring 
them  up  to  a  charged  electroscope.  There  will  be  no  change  in  the 
divergence  of  the  gold  leaf.  Now  test  the  rod  and  the  fur  separately  ; 
they  will  show  opposite  electrifications. 

This  experiment  shows  (1)  that  one  kind  of  electrification 
is  not  developed  without  the  other;  and  (2)  that  they  are 
developed  in  equal  quantities,  because  they  neutralize  each 
other  when  together. 

378.  Conductors  and  Nonconductors.  —  Fasten  a  smooth  metal- 
lic button  to  a  rod  of  sealing  wax  and  connect  the  button  with  the 
knob  of  the  electroscope  by  a  fine  copper  wire,  50  to  100  cm.  long. 
Hold  the  sealing  wax  in  the  hand  and  touch  the  button  with  an  elec- 
trified glass  tube.     The  divergence  of  the  leaf  indicates  that  it  is 
electrified.     If  we  repeat  the  experiment,  using  a  silk  thread  in  place 
of  the  wire,  no  such  effect  will  be  produced. 


ELECTRIFICATION  297 

All  substances  may  be  roughly  classed  under  two  heads, 
conductors  and  nonconductors.  In  the  former  if  one  point 
of  the  body  is  electrified  by  any  means,  the  electrification 
spreads  over  the  whole  body,  but  in  a  nonconductor  the 
electrification  is  confined  to  the  vicinity  of  the  point  where 
it  is  excited.  Nonconductors  are  commonly  called  insula- 
tors. Substances  differ  greatly  in  their  conductivity,  so 
that  it  is  not  possible  to  divide  them  sharply  into  two 
classes.  There  is  no  substance  that  is  a  perfect  con- 
ductor ;  neither  is  there  any  that  affords  perfect  in- 
sulation. Metals,  carbon,  and  the  solution  of  some  acids 
and  salts  are  the  best  conductors.  Among  the  best  insula- 
tors are  paraffin,  turpentine,  silk,  sealing  wax,  India  rubber, 
gutta-percha,  dry  glass,  porcelain,  mica,  shellac,  spun 
quartz  fibers,  and  liquid  oxygen.  Some  insulators,  like 
glass,  become  good  conductors  when  heated  to  a  semi-fluid 
condition. 

379.  Probable  Nature  of  Electrification.  —  It  was  sug- 
gested by  Faraday  that  the  electrification  of  a  body  is  a 
strained  condition  of  the  ether  which  surrounds  it  and 
pervades  it.  This  conclusion  is  supported  by  many  facts, 
such  as  action  at  a  distance,  rupture  of  bodies  by  over- 
charging, etc.  Conductors  differ  from  insulators  in  this  : 
in  the  former,  the  molecular  mobility  is  such  that  this 
state  of  strain  is  continually  giving  way,  while  in  the 
latter  considerable  distortion  is  possible  before  the  mole- 
cules yield  to  the  strain.  The  phenomena  of  attraction  and 
repulsion  exhibited  by  electrified  bodies  are  due  to  the  at- 
tempt of  the  strained  ether  in  and  around  the  bodies  to 
return  to  its  normal  condition.  In  producing  electrifica- 
tion, work  is  done  in  distorting  the  medium  ;  hence  elec- 
trification is  a  form  of  potential  energy. 


298 


ELECTROSTATICS 


II.  ELECTROSTATIC  INDUCTION 

380.  Electrification  by  Induction.  —  Rub  a  glass  tube  with  silk 
and  bring  it  near  the  top  of  an  electroscope.     The  leaves  begin  to 
diverge  when  the  tube  is  some  distance  from  the  knob  (Fig.  312)  and 

the  amount  of  divergence  increases 
as  the  tube  approaches.  When  the 
tube  is  removed  the  leaves  collapse. 

Since  the  leaves  do  not  re- 
main apart,  it  is  evident  that 
there  has  been  no  transfer  of 
electrification  from  the  tube 
to  the  electroscope.  The 
electrification  produced  in 
the  electroscope  when  the 
electrified  body  is  brought 
near  it  is  owing  to  electro- 
static induction.  That  such 
an  effect  should  occur  is 
easily  understood  when  we 
recall  Faraday's  view  of  electrification  as  a  distortion  of 
the  ether  about  the  body.  Evidently,  then,  any  body 
placed  within  this  electrical  field  should  be  electrified. 

381.  Sign  of  the  Induced  Charges.  —  Lay  a  smooth  metallic  ball 
on  a  dry  plate  of  glass.     Connect  it  with  the  knob  of  the  electroscope 
by  means  of  a  stout  wire  with  an 

insulating  handle  (Fig.  313).  The 
ball  and  the  electroscope  now  form 
one  continuous  conductor.  Bring 
near  the  ball  an  electrified  glass 
tube ;  the  leaves  of  the  electroscope 
diverge.  Before  withdrawing  the 


Fig.  312 


Fig.  313 


excited  tube,  remove  the  wire  conductor.  The  electroscope  remains 
charged,  and  it  will  be  found  to  be  positive.  A  similar  test  made 
of  the  ball  will  show  that  it  is  negatively  charged. 


ELECTROSTATIC  INDUCTION 


299 


Hence,  we  learn  that  when  an  electrified  body  is  brought 
near  an  object  it  induces  the  opposite  kind  of  electrification 
on  the  side  next  it  and  the  same  kind  on  the  remote  side. 

382.  Equality  of  the  Induced  Charges.  —  Put  a  metallic  vessel, 
like  the  one  shown  in  Fig.  315,  on  a  glass  plate  and  connect  it  with 
the  knob  of  an  electroscope  by  a  fine  wire.  Attach  a  silk  thread  to  a 
metallic  ball  about  an  inch  in  diameter,  and  charge  the  ball,  holding 
it  by  the  silk  thread.  Lower  the  charged  ball  into  the  vessel  and  ob- 
serve that  the  leaves  of  the  electroscope  diverge  as  the  ball  enters  the 
vessel.  The  divergence  increases  till  the  ball  has  been  lowered  per- 
haps two  inches  below  the  top,  and  then  remains  the  same,  even  when 
the  ball  touches  the  bottom  and  communicates  its  charge  to  the  in- 
sulated vessel. 

Suppose  the  ball  charged  positively;  it  induces  a  nega- 
tive charge  on  the  interior  of  the  vessel  and  repels  a  posi- 
tive charge  to  the  outside.  This  positive  charge  is  equal  to 
the  charge  on  the  ball,  for  the  divergence  of  the  leaves  does 
not  change  when  the  ball  gives  up  its  charge  to  the  vessel. 
The  charge  on  the  ball  neutralizes  the  equal  negative 
charge  on  the  interior, 
leaving  the  equal  positive 
charge  on  the  exterior. 

Discharge  the  electroscope 
and  charge  the  ball  a  second 
time.  After  it  has  been  low- 
ered into  the  insulated  vessel 
without  touching  it,  place  the 
finger  on  the  ball  of  the  electro- 
scope ;  the  leaves  will  collapse. 
Remove  the  finger  and  lift 
the  ball  by  the  silk  thread; 
the  leaves  will  again  diverge. 
Lower  the  ball  again  till  it  p.  ^iV 

touches    the   vessel,   and   the 

leaves  will  again   collapse.      The  charge  induced  on  the   inside   is 
exactly  neutralized  by  the  inducing  charge  on  the  ball. 


300 


ELECTROSTATICS 


Hence,  the  induced  and  the  inducing  charges  are  equal  to 
each  other. 

383.  Charging  an  Electroscope  by  Induction.  —  Hold  a  finger 
on  the  ball  of  the  electroscope  and  bring  near  it  an  electrified  glass 
tube  (Fig.  314).  Remove  the  finger  before  taking  away  the  tube; 
the  electroscope  will  be  charged  negatively.  If  a  stick  of  electrified 
sealing  wax  be  used  instead  of  the  glass  tube,  the  electroscope  will 
be  charged  positively. 


III.  ELECTRICAL  DISTRIBUTION 

384.   The  Charge  on  the  Outside  of  a  Conductor.  —  Place  a 
round  metallic  vessel  of  about  one  liter  capacity  on  an  insulated 

support  (Fig.  315).  Electrify 
strongly  and  test  in  succession 
both  the  inner  and  the  outer 
surface,  using  a  proof  plane  to 
convey  the  charge  to  the  elec- 
troscope. The  inner  surface 
will  give  no  sign  of  electri- 
fication. 

Hence,  it  appears  that 
the  electrical  charge  of  a 
conductor  is  confined  to  its 
outer  surface. 

385.   Effect   of    Shape. - 

Charge  electrically  an  insulated 

egg-shaped  conductor  (Fig.  316). 

Touch  the  proof 
Fig.  3 1 5  plane  to  the  large 

end,  and  convey 

the  charge  to  the  electroscope.  Notice  the  amount 
of  divergence  of  the  leaves.  Test  the  side  and  the 
small  end  of  the  conductor  in  the  same  way.  The 
greatest  divergence  of  the  leaves  will  be  produced  by  the 
charge  from  the  small  end  and  the  least  from  the  sides.  Fig.  316 


ELECTRICAL  DISTRIBUTION  301 

The  experiment  shows  that  the  surface  density  is 
greatest  at  the  small  end  of  the  conductor. 

By  surface  density  is  meant  the  quantity  of  electrification 
on  a  unit  area  of  the  surface  of  the  conductor. 

The  distribution  of  the  charge  is,  therefore,  affected  by  the 
shape  of  the  conductor,  the  surface  density  being  greater  the 
greater  the  curvature. 

386.  Action  of  Points.  —  Attach  a  sharp-pointed  rod  to  one  pole 
of  an  electrical  machine  (§  399),  and  suspend  two  pith  balls  from  the 
same  pole.  When  the  machine  is  worked  there  will  be  little  or  no 
separation  of  the  pith  balls.  Hold  a  lighted  candle  near  the  pointed 
rod ;  the  candle  flame  will  be  blown  away  as 
by  a  stiff  breeze  (Fig.  317). 

The  experiment  shows  that  an  elec- 
tric charge  is  carried  off  by  pointed 
conductors.  This  conclusion  might 
have  been  drawn  from  the  preceding 
experiment.  When  the  curvature  of 
the  egg-shaped  conductor  becomes 
very  great  so  that  the  surface  becomes  Fjg  317 

pointed,  the  surface  density  also  be- 
comes great  and  there  is  an  intense  field  of  electric 
force  in  the  immediate  neighborhood.  The  air  particles 
touching  the  point  become  heavily  charged  and  are  then 
repelled;  other  particles  take  their  place  and  are  in  turn 
repelled  and  form  an  electrical  wind.  The  conductor  gives 
up  its  charge  to  the  repelled  particles  of  air. 

Questions 

1.  What  would  be  the  effect  of  replacing  the  ball  of  an  electro- 
scope by  a  sharp  point? 

2.  Why  is  the  gold  leaf  of  an  electroscope  inclosed  in  a  case? 

3.  What  kind  of  a  handle  must  a  proof  plane  have  ?     Why  ? 


302  ELECTROSTATICS 

4.  Why  must  electrical  apparatus  be  thoroughly  dry  to  work 
well? 

5.  If  a  long  silk  ribbon  is  doubled  and  then  stroked  between  the 
folds  of  a  piece  of  fur,  the  two  halves  will  repel  each  other.     Why  ? 

6.  Why  does  not  a  metal  rod  held  in  the  hand  and  rubbed  with 
silk  show  electrification  ?     Can  it  be  made  to  do  so,  and  how  ? 

7.  Place  an  electroscope  in  a  cage  of  wire  netting.     Why  is  it 
not  affected  by  an  electrified  glass  rod  held  near  it  ? 

8.  How  is  it  possible  to  electrify  an  electroscope,  using  only  a 
silk  handkerchief? 

9.  If  electrification  is  a  form  of  energy,  whence  its  source  in  the 
case  of  an  electrified  glass  rod  ? 

10.  If  a  pith  ball  were  separated  from  an  electrified  glass  rod  by 
a  perfect  vacuum,  would  there  be  any  attraction? 

IV.    ELECTRIC  POTENTIAL  AND  CAPACITY 

387.  The  Unit  of  Electrification  or  Charge.  —  Imagine  two 
minute    bodies  similarly  charged  with    equal   quantities 
of  electricity.     They  will  repel  each  other.     If  the  two 
equal  and  similar  charges  are  one  centimeter  apart  in  air, 
and  if   they  repel  each  other  with  a  force  of  one  dyne, 
then  the  charges  are  both  unity.     The  electrostatic  unit  of 
quantity  is  that  quantity  which  will  repel  an  equal  and  similar 
quantity  at  a  distance  of  one  centimeter  in  air  with  a  force  of 
one  dyne.     It  is  necessary  to  say  "  in  air"  because,  as  will 
be  seen  later,  the  force  between  two  charged  bodies  depends 
on  the  nature  of  the  medium  between  them  (§  393). 

This  electrostatic  unit  is  very  small  and  has  no  name. 
In  practice,  a  larger  unit,  called  the  coulomb,  is  employed. 
It  is  equal  to  3  x  109  electrostatic  units. 

388.  Potential  Difference. — The  analogy  'between  pres- 
sure in  hydrostatics  and  potential  in  electrostatics  is  a  very 
convenient  and  helpful  one.     Water  will  flow  from  ,the 


ELECTRIC  POTENTIAL  AND   CAPACITY 


303 


n 


tank  A  to  the  tank  B  (Fig.  318)  when  the  stopcock 
the  connecting  pipe  is  open  if  the  hydrostatic  pressure  at 
a  is  greater  than  at  6;  and  the  flow 
is  attributed  directly  to  this  differ- 
ence of  pressure. 

In  the  same  way,  if  there  is  a 
flow  of  positive  electricity  from  A 
to  B  when  the  two  conductors  are 
connected  by  a  conducting  wire  r 
(Fig.  319),  the  electrical  potential 
is  said  to  be  higher  at  A  than  at 
J5,  and  the  difference  of  electrical 
potential  between  A  and  B  is  as- 
signed as  the  cause  of  the  flow.  In  both  cases  the  flow 
is  in  the  direction  of  the  difference  of  pressure  or  dif- 
ference of  potential,  irre- 
spective of  the  fact  that  B 
may  already  contain  more 
water  because  of  its  large 
cross  section,  or  a  greater 


Fig.  318 


Fig.  319 


electric  charge  because  of  its  larger  capacity. 

If  the  electric  charges  in  a  system  of  connected  conduc- 
tors is  in  a  stationary  or  static  condition,  there  is  then  no 
potential  difference  between  different  points  of  the  system. 
f  The  potential  difference  between  two  conductors  is  meas- 
mred  by  the  work  done  in  carrying  a  unit  electric  charge 
vfrom  the  one  to  the  other. 


389.  Unit  Potential  Difference.  —  There  is  unit  potential 
difference  between  two  conductors  when  one  erg  of  work 
is  required  to  transfer  the  unit  electric  charge  from  the 
one  conductor  to  the  other.  This  is  called  the  absolute 
unit ;  for  practical  purposes  it  has  been  found  more  con- 


304  ELECTROSTATICS 

venient  to  employ  a  unit  of  potential  difference  (P.D.), 
which  is  g-J-Q  of  the  absolute  unit,  and  which  is  called  the 
volt,  in  honor  of  the  Italian  physicist,  Alessandro  Volta. 

390.  Zero  Potential.  — In  measuring  the  potential  differ- 
ence between  a  conductor  and  the  earth,  the  potential  of 
the  earth  is  assumed  to  be  zero.     The  potential  difference 
is  then  numerically  the  potential  of  the  conductor.     If  a 
conductor  of   positive  potential  be   connected   with   the 
earth  by  an  electric  conductor,  the  positive  charge  will 
flow  to  the  earth.     If  the  conductor  has  a  negative  poten- 
tial, the  flow  of  the  positive  quantity  will  be  in  the  other 
direction. 

391.  Electrostatic  Capacity. — If  water  be  poured  into  a 
cylindrical  jar  until  it  is  10  cm.  deep,  the  pressure  on  the 
bottom  of  the  jar  is  10  gm.  of  force  per   square    centi- 
meter.    If  the  depth  of  the  water  be  increased  to  20  cm., 
the   pressure    will   be  20  gm.  of  force  per  square  centi- 
meter (§  48).     It  thus  appears  that  there  is  a  constant 
relation  between  the  quantity  of  water  Q  and  the  pressure 

P\  that  is,  -^  =  (7,  a  constant. 

Again,  if  a  gas  tank  be  filled  with  gas  at  atmospheric 
pressure,  it  will  exert  a  pressure  of  1033  gm.  of  force  per 
cm2,,  (§  68).  If  twice  as  much  gas  be  pumped  into  the 
tank,  the  pressure  by  Boyle's  law  (§  74)  will  be  doubled 
at  the  same  temperature ;  that  is,  there  is  a  constant  re- 
lation between  the  quantity  of  gas  Q  and  the  pressure  P 

of  the  gas  in  the  tank,  or  -^  =  (7,  a  constant  as  before. 

In  the  same  way,  if  an  electric  charge  be  given  to  an 
insulated  conductor,  its  potential  will  be  raised  above  that 
of  the  earth.  If  the  charge  be  doubled,  the  potential 


Alessandro  Volta  (1748- 
1827)  was  born  at  Como,  It- 
aly. He  was  professor  of 
physics  at  the  University  of 
Pavia,  and  was  noted  for  his 
researches  and  investigations 
in  electricity.  The  voltaic 
cell,  the  electroscope,  the 
electrical  condenser,  and  the 
electrophorus  are  due  to  his 
genius. 


Georg  Simon  Ohm  ( 1 789- 
1854)  was  born  in  Erlangen, 
Bavaria,  and  was  educated 
at  the  University  of  that 
town.  He  began  his  inves- 
tigations by  measuring  the 
electrical  conductivity  of 
metals.  In  1827  he  an- 
nounced the  electrical  law 
named  in  his  honor,  and  in 
1842  he  was  elected  to  a  pro- 
fessorship in  the  University 
of  Munich. 


ELECTRIC  POTENTIAL  AND   CAPACITY  305 

difference  between  the  conductor  and  the  earth  will  also 
be  doubled.  Precisely  as  in  the  case  of  the  water  and  of 
the  gas,  there  is  a  constant  relation  between  the  amount 
of  the  charge  Q  and  the  potential  difference  V  between 

the  conductor  and  the  earth ;  that  is,  j-=  0.     This  ratio 

or  constant  O  is  the  electrostatic  capacity  of  the  conductor. 
If   V=  1,  then  O  =  Q ;  from  which  it  follows  that  the 
electrostatic  capacity  of  a  conductor  is  equal  to  the  charge  re- 
quired to  raise  its  potential  from  zero  to  unity. 

From  ^=  0 we  have<?  =  CVt  and  F=  ^  (Equation  34.) 

392.  Condensers.  —  Support  a  metal  plate  in  a  vertical  position 
on  an  insulating  base  (Fig.  320.)  Connect  it  to  the  knob  of  an  elec- 
troscope by  a  fine  copper  wire.  Charge  the  plate  until  the  leaves  of 
the  electroscope  show  a  wide  divergence.  Now  bring  an  uninsulated 
conducting  plate  near  the  charged  one 
and  parallel  to  it.  The  divergence  of 

the  leaves  will  decrease ;  remove  the  un-  jf|      jf|    !       >6cope 

insulated  plate,  and  the  divergence  will 
increase  again. 

The  capacity  of  an  insulated 
conductor  is  not  dependent  on  its 
dimensions  alone,  but  it  is  in-  Fig.  320 

creased  by  the  presence  of  another 

conductor  connected  with  the  earth.  The  effect  of  this 
latter  conductor  is  to  decrease  the  potential  to  which  a 
given  charge  will  raise  the  insulated  one.  Such  an  ar- 
rangement of  parallel  conductors  separated  by  an  insulator 
or  dielectric  is  called  a  condenser. 

A  condenser  is  a  device  which  greatly  increases  the 
charge  on  a  conductor  without  increasing  its  potential. 
In  other  words,  the  plate  connected  with  the  earth  greatly 
increases  the  capacity  of  the  insulated  conductor. 


306 


ELECTROS  TA  TICS 


393.  Influence  of  the  Dielectric.  —  Charge  the  apparatus  of  the 
last  experiment,  with  the  uninsulated  plate  at  a  distance  of  about  5 
cm.  from  the  charged  plate  and  parallel  to  it,  thrust  suddenly  be- 
tween the  two  a  cake  of  clean  paraffin  as  large  as  the  metal  plates  or 
larger,  and  from  2  to  4  cm.  thick.  Note  that  the  leaf  of  the  electro- 
scope (Fig.  310)  collapses  slightly.  Remove  the  paraffin  quickly,  and 
the  divergence  will  increase  again.  A  cake  of  sulphur  will  produce 
a  more  marked  effect  on  the  divergence  of  the  leaf. 

The  presence  of  the  paraffin  or  the  sulphur  increases 
the  capacity  of  the  condenser  and,  hence,  decreases  its 
potential,  the  charge  remaining  the  same.  Paraffin  and 
sulphur,  as  examples  of  dielectrics,  are  said  to  have  a 
larger  dielectric  capacity  or  dielectric 
constant  than  air.  Glass  has  a  dielec- 
tric capacity  from  four  to  ten  times 
greater  than  air. 


Fig.  321 


394.  The  Leyden  Jar  is  a  common  and 
convenient  form  of  condenser.  It  con- 
sists of  a  glass  jar  coated  part  way  up, 
both  inside  and  outside,  with  tin-foil 
(Fig.  321).  Through  the  wooden  or 
ebonite  stopper  passes  a  brass  rod,  ter- 
minating on  the  outside  in  a  ball  and  on 
the  inside  in  a  metallic  chain  which  reaches  the  bottom  of 
the  jar.  The  glass  is  the  dielec- 
tric separating  the  two  tin-foil 
conducting  surfaces. 


395.  Charging  and  Discharg- 
ing a  Jar.  —  To  charge  a  Ley- 
den  jar  connect  the  outer 
surface  to  one  pole  of  an  electri- 
cal machine  (§  399),  either  by 


Fig.  322 


ELECTRIC  POTENTIAL  AND   CAPACITY 


307 


a  metallic  conductor  or  by  holding  the  jar  in  the  hand. 
Hold  the  ball  against  the  other  pole.  To  discharge  a 
Leyden  jar  bend  a  wire  into  the  form  of  the  letter  V. 
With  one  end  of  the  wire  touching  the  outer  surface  of  the 
jar  (Fig.  322),  bring  the  other  around  near  the  ball,  and 
the  discharge  will  take  place. 

396.  Seat  of  Charge.  —  Charge  a  Leyden  jar  made  with  movable 
metallic  coatings  (Fig.  323)  and  set  it  on  an  insulating  stand.  Lift 
out  the  inner  coating,  and  then,  taking  the  top 
of  the  glass  vessel  in  one  hand,  remove  the 
outer  coating  with  the  other.  The  coatings 
now  exhibit  no  sign  of  electrification.  Bring 
the  glass  vessel  near  a  pile  of  pith  balls ;  they 
will  be  attracted  to  it,  showing  that  the  glass 
is  electrified.  Reach  over  the  rim  with  the 
thumb  and  forefinger  and  touch  the  glass.  A 
slight  discharge  may  be  heard.  Now  build  up 
the  jar  by  putting  the  parts  together;  the  jar 
will  still  be  highly  electrified  and  may  be  dis- 
charged in  the  usual  way. 


Fig.  323 


This  experiment  was  devised  by 
Franklin ;  it  shows  that  electrification 
is  a  phenomenon  of  the  glass,  and  that  the  metallic  coat- 
ings serve  merely  as^  conductors,  making  it  possible  to 
discharge  all  parts  of  the  glass  at  once. 

397.  Theory  of  the  Leyden  Jar.  —  A  Leyden  jar  may  be 
broken  by  over  charging,  may  be  discharged  by  heating, 
and  if  heavily  charged  is  not  completely  discharged  by 
connecting  the  two  coatings ;  if  left  standing  a  few  sec- 
onds, the  two  coatings  gradually  acquire  a  small  potential 
difference  and  a  second  small  discharge  may  be  obtained, 
known  as  the  residual  charge.  It  appears,  therefore,  that 
the  glass  of  a  charged  jar  is  strained  or  distorted ;  like  a 


308  ELECTROSTATICS 

twisted  glass  fiber,  it  does  not  return  at  once  to  its  normal 
state  when  released. 

The  two  surfaces  of  the  glass  are  oppositely  electrified, 
the  one  charge  acting  inductively  through  the  glass  and 
producing  the  opposite  electrification  on  the  other  surface. 
The  two  charges  are  held  inductively  and  are  said  to  be 
"bound,"  in  distinction  from  the  charge  on  an  insulated 
conductor,  which  is  said  to  be  "free." 

Questions 

1.  Why  is  not  a  Leyden  jar  standing  on  a  cake  of  paraffin  dis- 
charged by  touching  the  ball  with  the  finger? 

2.  Why  is  the  capacity  of  a  Leyden  jar  increased  by  connecting  its 
outer  surface  with  the  earth?     Is  the  result  the  same  if  one  coating 
is  connected  to  one  pole  of  an  electrical  machine  and  the  other  coat- 
ing to  the  other  pole? 

3.  Why  will  a  dust-covered  electrified  body  soon  lose  its  charge? 

4.  Cuneus  tried  to  charge  a  bowl  of  water  by  holding  it  in  his 
hand  while  a  chain  from  an  electrical  machine  dipped  into  it.     When 
he  lifted  out  the  chain  he  got  a  severe  shock.     Why? 

5.  Why  is  it  unsafe'to  touch  the  ball  first  when  discharging  a  Ley- 
den jar  by  means  of  a  bent  wire?     Would  it  make  any  difference  if 
the  wire  were  held  by  an  insulating  handle? 

6.  When  a  charged  proof  plane  touches  the  ball  of  an  electroscope 
does  it  give  up  all  its  charge  ? 

7.  When  a  charged  proof  plane  touches  the  inside  of  a  hollow 
insulated  conductor,  does  it  give  up  all  its  charge?    Why? 

8.  If  a  condenser  whose  capacity  is  200  c.  g.  s.  units  is  charged 
with  2000  c.g.s.  units  of  electricity,  what  is  the  potential  difference 
in  volts  between  its  two  coatings  (§  391)? 

V.    ELECTRICAL  MACHINES 

398.  The  Electrophorus.  —  The  simplest  induction  elec- 
trical machine  is  the  electrophorus  (Fig.  324),  invented 
by  Volta.  A  cake  of  resin  or  disk  of  vulcanite  (yi)  rests 


ELECTRICAL   MACHINES 


309 


in  a  metallic  base  (1?).  Another  metallic  disk  or  cover 
((7)  is  provided  with  an  insulating  handle  (D).  The  resin 
or  vulcanite  is  electrified  by  rubbing  with  dry  flannel  or 
striking  with  a  catskin,  and  the  metal  disk  is  then  placed 
on  it.  Since  the  cover  touches  the  nonconducting  resin 
or  vulcanite  (A)  in  a  few 
points  only,  the  negative 
charge  due  to  the  friction  is 
not  removed.  The  two  disks 
with  the  film  of  air  between 
them  form  a  condenser  (§  392) 
of  great  capacity. 

Touch  the  cover  momentarily 
with  the  finger,  and  the  repelled 
negative  charge  passes  to  the  earth,  P.  324 

leaving   the  cover  at  zero  poten- 
tial.    Lift  it  by  the  insulating  handle,  the  positive  charge  becomes 
free  (§  397),  and  a  spark  may  be  drawn  Jby  holding  the  finger  near  it. 
This  operation  may  be  repeated  an  indefinite  number  of  times  with- 
out sensibly  reducing  the  charge  on  the  vulcanite. 

When  the  cover  is  lifted  by  the  insulating  handle,  work 
is  done  against  the  electrical  attraction  between  the  nega- 
tive charge  on  the  vulcanite  and  the  positive  on  the  cover. 
The  energy  of  the  charged  cover  represents  this  work. 
The  electrophorus  is,  therefore,  a  device  for  the  continuous 
transformation  of  mechanical  work  into  the  energy  of  electric 
charges. 

399.  Influence  Electrical  Machines. — There  are  many 
influence  or  induction  electrical  machines,  but  it  will 
suffice  to  describe  only  one  as  the  principle  is  always  the 
same. 

The  Holtz  machine,  as  modified  by  Toepler  and  Voss, 
is  illustrated  in  Fig.  325.  There  are  two  glass  plates,  e' 


310 


ELECTROSTATICS 


and  e,  about  5  mm.  apart,  the  former  stationary  and  the 
latter  turning  about  an  insulated  axle  by  means  of  the 
crank  h  and  a  belt.  The  stationary  plate  supports  at 


Fig.  325 

the  back  two  paper  sectors,  c  and  <?',  called  armatures. 
Between  them  and  the  stationary  plate  e1  are  disks  of  tin- 
foil connected  by  a  narrow  strip  of  the  same  material. 
The  disks  are  electrically  connected  with  two  bent  metal 
arms,  a  and  a1  (opposite  a),  which  carry  at  the  other  end 
tinsel  brushes  long  enough  to  rub  against  low  brass  but- 
tons cemented  to  small  tin-foil  disks,  called  carriers,  on 
the  front  of  the  revolving  plate.  Opposite  the  paper  sec- 
tors and  facing  them  are  two  metal  rods  with  several 
sharp-pointed  teeth  set  close  to  the  revolving  plate,  but 
not  touching  the  metal  buttons  and  carriers.  The  diag- 
onal neutralizing  rod  d  has  tinsel  brushes  in  addition 
to  the  sharp  points.  The  two  insulated  conductors,  ter- 
minating in  the  balls,  m  and  n,  have  their  capacity  in- 
creased by  connection  with  the  inner  coating  of  two  small 


ELECTRICAL  MACHINES  311 

Leyden  jars,  i  and  i'  ;  the  outer  coatings  are  connected 
under  the  base  of  the  machine. 

400.  Action  of  the  Machine.  —  There  is  usually  enough 
excitation  due  to  friction  or  to  the  contact  of  dissimilar 
substances  to  furnish  the  very  slight  initial  charge  on  the 
armatures.      These  small  charges   are  necessary  to   the 
operation  of  all  induction  machines.     Suppose  the  two 
armatures,  c  and  cr  (Fig.  325),  slightly  charged,  the  one 
on  the  right  positive.     The  armatures  act  by  induction  on 
the  horizontal  conductors  opposite  them ;   negative  elec- 
tricity escapes  from  the  points  on  the  right  hand  "  comb  " 
to  the  revolving  plate,  and  positive  from  the  left.     The 
brushes  on  the  neutralizing  rod  d  are  set  so  as  to  connect 
two  carriers  at  opposite  ends  of  the  rod  just  before  these 
carriers  pass   beyond   the   influence  of   the  armatures  c 
and  c1.     The  carriers  connected  by  the  rod  d  thus  acquire 
by  induction  positive  and  negative  charges  respectively, 
which  they  carry  forward  as  the  plate  revolves  until  they 
are  brought  into  momentary  connection  with  the  arma- 
tures by  means  of  the  brushes  and  the  bent  conductors  a 
and  a'.     The  carriers  then  deliver  their  small  charges  to 
the  armatures.     At  the  same  time  the  revolving  plate 
carries  around  the  charges  escaping  by  the  points  and 
gives  them  up,  at  least  in  part,  to  the  conductors,  a  and  a'. 
The  potential  difference  between  c  and  c'  is  thus  increased 
by  induction  and  the  charges  carried  by  the  revolving 
plate  until  a  discharge  takes  place  between  the  balls  m 
and  7i,  when  they  are  separated. 

401.  Experiments  with  Electrical  Machines. —  1.  Attraction 

and  repulsion.  Place  a  number  of  bits  of  paper  on  the  cover  of  a 
charged  electrophorus.  Lift  it  by  the  insulating  handle.  The 
charged  pieces  of  paper  fly  off  the  plate. 


312 


ELECTROSTATICS 


Fig.  326 


Three  bells  are  suspended  from    a   metal   bar  (Fig.  326).     The 
middle  one  is  insulated  from  the  bar ;  the  others  are  suspended  by 

chains.  Connect  the  bar  to  one  pole  of 
an  electrical  machine  and  the  middle 
bell  to  the  other.  The  small  brass  balls 
between  the  bells  are  suspended  by  silk 
cords ;  they  swing  to  and  fro  between  the 
bells,  carrying  positive  charges  in  one 
direction  and  negative  in  the  other.  This 
apparatus,  called  the  electrical  chimes, 
is  of  interest  because  it  was  employed  by 
Franklin  in  his  lightning  experiments  to 
announce  the  electrification  of  the  cord 
leading  to  the  kite  (§  402). 

2.  Discharge  by  points.  Connect  an 
electrical  tournique  (Fig.  327)  to  one  of 
the  conductors  of  an  electrical  machine, 
the  other  conductor  being  grounded.  When  the  machine  is  turned, 
the  whirl  rotates  rapidly  (§  386). 

3.  Mechanical  effects.  Hold  a  piece  of 
cardboard  between  the  discharge  balls  of  an 
electrical  machine.  It  will  be  perforated  by 
a  spark  and  the  holes  will  be  burred  out 
on  both  sides.  A  thin  dry  glass  plate,  or  a 
thin  test  tube  over  a  sharp  point,  may  be  per- 
forated by  a  heavy  discharge. 
•  4.  Heating  effects.  Charge  a  Leyden  jar 
and  connect  its  outer  coating  with  a  gas 
burner  by  a  chain  or  wire.  Turn  on  the 
gas  and  bring  the  ball  of  the  jar  near  enough  to  the  opening  in  the 
burner  to  allow  a  spark  to  pass.  The  gas  will  be  lighted  by  the  dis- 
charge. 

Fill  a  gas  pistol  (Fig. 
328)  with  a  mixture  of  coal 
gas  and  air.  Discharge  a 
Leyden  jar  through  the  mix- 
ture. It  will  explode  and  the 
cork  or  ball  will  be  shot  out 
Fig.  328  with  some  violence. 


Fig.  327 


ATMOSPHERIC  ELECTRICITY 


313 


5.  Magnetic  effects.  —  Wind  insulated 
copper  wire  around  a  small  glass  tube 
(Fig.  329),  and  place  inside  the  tube  a 
piece  of  darning  needle.  Discharge  a 
Leyden  jar  through  the  wire.  The 
needle  will  be  magnetized.  A  similar 
eifect  may  be  produced  by  placing  a 
large  sewing  needle  across  a  strip  of 
tin-foil  forming  a  part  of  the  discharge 
circuit  of  a  Leyden  jar. 


Q 


Fig.  329 


VI.   ATMOSPHERIC  ELECTRICITY 

402.  Lightning. — Franklin  demonstrated   in  1752  that 
lightning  is  identical  with  the  electric  spark.     He  sent  up 
a  kite  during  a  passing  storm,  and  found  that  as  soon  as 
the   hempen  string   became  wet,  long   sparks   could   be 
drawn  from  a  key  attached  to  it,  Leyden  jars  could  be 
charged,  and  other  effects  characteristic  of  static  electrifi- 
cation could  be  produced. 

Lightning  flashes  are  discharges  between  oppositely 
charged  bodies.  They  occur  either  between  two  clouds 
or  between  a  cloud  and  the  earth.  The  rise  of  potential 
in  a  cloud  causes  a  charge  to  accumulate  on  the  earth 
beneath  it.  If  the  stress  in  the  air  reaches  a  value  of 
about  400  dynes  per  cm.2,  the  air  breaks  down,  or  is 
ruptured,  like  any  other  dielectric,  and  the  two  opposite 
charges  unite  in  a  long  zigzag  flash.  A  lightning  flash 
allows  the  strained  medium  to  return  to  equilibrium. 
The  coming  together  of  the  air  surfaces,  which  are  sepa- 
rated in  the  rupture,  produces  a  violent  crash  of  thunder. 

403.  The  Lightning  Rod.  —  Support  two  round  metal  plates, 
T  and   T,  one  above  the  other  and  a  few  centimeters  apart  (Fig. 
330).     The  upper  plate  must  be  carefully  insulated  except  from  the 


814  ELECTROSTATICS 

pole  of  the  electrical  machine  and  the  inner  coating  of  a  Ley  den  jar 

L.     Two  of  the  short  rods  on  the  lower  plate  terminate  in  small 

balls ;  the  other  and  shortest  one  is  pointed.     When  the  machine  is 

II  II  worked,  the  tension  between 

A  the  plates  increases,  but  it  is 

difficult    to    make   a   spark 
pass;    if   one  does  pass,  it 
will     strike      the     pointed 
,         rod. 

The  experiment  il- 
lustrates the  protection 

P.  3-,c  afforded  by  a  pointed 

conductor. 

A  lightning  rod  should  conform  to  the  following 
requirements  : 

First.  It  should  be  perfectly  continuous,  of  sufficient 
size  to  resist  fusion,  and  made  preferably  of  strands  of 
wire  twisted  together  as  a  cable.  Iron  cables  are  as  good 
as  copper  ones. 

Second.  The  upper  end  should  terminate  in  points  and 
should  be  higher  than  adjacent  parts  of  the  building. 
The  lower  end  should  pass  down  into  the  earth  until  it 
enters  a  moist  conducting  stratum. 

Third.  The  rod  should  be  fastened  to  the  building 
without  insulators,  and  all  metal  parts  of  the  roof  should 
be  connected  with  the  main  conductor.  It  is  better  to 
have  two  or  three  descending  rods  than  one,  and  all  the 
points  and  rods  should  be  connected  together  as  a  net- 
work. 

404.  Oscillatory  Discharge.  —  When  a  Leyden  jar  is  highly 
charged,  the  potential  difference  between  its  coatings  increases  until 
the  dielectric  between  the  discharge  terminals  suddenly  breaks  down 
and  a  spark  passes.  This  discharge  usually  consists  of  several  oscil- 
lations or  to-and-fro  discharges,  like  the  vibrations  of  an  elastic 


Benjamin  Franklin  (1706-1790)  was  born  at  Boston,  Massa- 
chusetts. In  his  twentieth  year  he  was  apprenticed  to  his  elder 
brother  in  the  printing  business.  When  forty  years  of  age  he  saw 
some  electrical  experiments  performed  with  a  glass  tube.  These 
excited  his  curiosity  and  he  began  experimenting  for  himself.  In 
less  than  a  year  he  had  discovered  the  discharging  effects  of  points 
and  worked  out  a  theory  of  electricity,  known  as  the  "one-fluid 
theory."  He  explained  the  charged  Leyden  jar,  established  the 
identity  of  lightning  and  the  electric  spark,  invented  an  electric 
machine,  and  introduced  the  lightning  rod  as  a  protection  against 
lightning.  He  was  distinguished  as  a  statesman,  diplomatist,  and 
scientist.  He  founded  the  American  Philosophical  Society  and 
the  University  of  Pennsylvania. 


ATMOSPHERIC  ELECTRICITY  315 

system,  or  the  surges  of  a  mass  of  water  after  sudden  release  from 
pressure.  Imagine  a  tank  with  a  partition  across  the  middle  arid 
filled  on  one  side  with  water.  If  a  small  hole  be  made  in  the  par- 
tition near  the  bottom,  the  water  will  slowly  reach  the  same  level  on 
both  sides  without  agitation;  but  if  the  partition  be  suddenly 
removed,  the  first  violent  subsidence  will  be  succeeded  by  a  return 
surge,  and  the  to-and-fro  motion  of  the  water  will  continue  with 
decreasing  violence  until  the  energy  is  all  expended. 

A  series  of  similar  surges  occurs  when  a  condenser  is  suddenly  dis- 
charged by  the  breaking  down  of  the  dielectric.  The  oscillatory 
character  of  such  electric  discharges  was  discovered  by  Joseph  Henry 
in  1842.  Its  importance  has  been  recognized  only  in  recent  times. 
Similar  electric  oscillations  probably  take  place  in  some  lightning 
flashes. 

405.  The  Aurora. — The  aurora  is  due  to  silent  dis- 
charges in  the  upper  regions  of  the  atmosphere.;  Within 
the  arctic  circle  it  occurs  almost  nightly,  and  sometimes 
with  indescribable  splendor.  The  illumination  of  the 
aurora  is  due  to  positive  discharges  passing  from  the 
higher  regions  of  the  atmosphere  to  the  earth.  In  our 
latitude  these  silent  streamers  in  the  atmosphere  are 
infrequent.  When  they  do  occur  they  are  accompanied 
by  great  disturbances  of  the  earth's  magnetism  and  by 
earth  currents.  Such  magnetic  disturbances  sometimes 
occur  at  the  same  time  in  widely  separated  portions  of  the 
earth. 


CHAPTER   XII 

ELECTRIC  CURRENTS 

I.  VOLTAIC  CELLS 

406.  An  Electric  Current.  —  When  a  condenser  is  dis- 
charged through  a  wire,  there  is  produced  in  and  around  I 
the  wire  a  state  called  an  electric  current.  Electrification 
is  a  condition  of  strain'in  the  dielectric  ;  the  electric  cur- 
rent rapidly  relieves  this  strain  through  the  discharging 
conductor.  If  the  state  of  strain  is  reproduced  by  the 
"generator"  as  fast  as  it  is  relieved  by  the  conductor,  the 
result  is  a  continuous  current.  ^To  accomplish  this  result 
work  must  be  done,  and  therefore  an  electric  current 
represents  energy.  The  expression  "  current  of  electric- 
ity," was  introduced  when  electricity  was  regarded  as  a 
fluid  which  flowed  from  higher  to  lower  potential  through 
a  wire,  just  as  water  flows  from  a  higher  to  a  lower  level 
through  a  pipe. 

To  produce  a  continuous  electric  current  through  a  con- 
ductor a  potential  difference  must  be  maintained  between 
its  terminals.      One  of  the   simplest 
means  of  doing  this  is  the  primary  or 
voltaic  cell. 


407.  The  Voltaic  Cell.  —  Support  a 
heavy  strip  of  zinc  and  one  of  sheet  copper 
(Fig.  331)  in  dilute  sulphuric  acid  (one  part 
acid  to  twenty  of  water).  After  the  zinc 
has  been  in  the  acid  a  short  time,  it  should  be 
Fig.  331  amalgamated  by  rubbing  it  with  mercury. 

316 


VOLTAIC  CELLS 


317 


There  will  be  no  apparent  change  when  the  plates  are  replaced  in  the 
acid,  until  the  two  are  connected  with  a  copper  wire;  a  multitude  of 
bubbles  of  hydrogen  gas  will  then  immediately  be  given  off  at  the  sur- 
face of  the  copper  plate.  The  action  ceases  as  soon  as  the  wires  are 
disconnected.  If  the  action  is  continued  for  some  time,  the  zinc  will 
waste  away,  while  the  copper  is  not  affected. 

Such  a  combination  of  two  conductors,  immersed  in  a 
compound  liquid,  called  an  electrolyte,  which  is  capable  of 
reacting  chemically  with  one  of  the  conductors,  is  called  a 
voltaic  cell.  The  name  is  derived  from  Volta  of  Padua, 
who  first  described  such  a  cell  in  1800. 

408.  Plates  Electrically  Charged.  —  In  a  condensing  electro- 
scope the  ball  at  the  top  is  replaced  by  a  brass  disk  coated  with  thin 
shellac  varnish  as  an  insulator.  Resting  on  it  is  a  second  disk  to 
which  is  fitted  an  insulating  handle.  The 
two  disks  with  the  shellac  varnish  between 
them  form  a  condenser  of  considerable 
capacity. 

Connect  the  wire  leading  from  the  copper 
plate  of  two  or  three  voltaic  cells  C  in  series 
(§  437)  to  the  lower  disk  A  of  the  electro- 
scope, and  the  wire  from  the  zinc  plate  to 
the  upper  disk  B  (Fig.  332).  Disconnect 
the  wires,  handling  them  one  at  a  time  by 
means  of  a  good  insulator  so  as  not  to  dis- 
charge the  condenser,  and  then  lift  the  top 
disk.  The  leaf  L  of  the  electroscope  will 
diverge,  and  a  test  with  an  electrified  glass 
rod  will  show  that  the  electroscope  is  charged 
positively..  This  positive  charge  was  derived 


B 

~-—  J 

=c~ 

A 
\ 

.  r\ 

E 

L 

V 

t^ 

ncr 

r\  rr 

from  the  copper  strip  of  the  cell.  Repeat  the  experiment  with  the 
zinc  plate  connected  to  the  lower  disk;  the  result  will  be  a  negative 
charge  on  the  gold  leaf. 

It  is  clear  from  this  experiment  that  the  plates  of  a  vol- 
taic cell  and  the  ivires  leading  from  them  are  electrically 
charged,  the  copper  positively  and  the  zinc  negatively.  The 


318  ELECTRIC  CURRENTS 

conducting  rods,  plates,  or  cylinders  in  a  voltaic  cell  are 
called  electrodes,  the  copper  the  positive  electrode  or  cathode 
and  the  zinc  the  negative  electrode  or  anode.  The  electric 
current  leaves  the  electrolyte  by  the  cathode  (meaning  the 
way  out)  and  enters  it  by  the  anode  (meaning  the  way  in). 

409.  The  Circuit.  —  The  circuit  of  a  voltaic  cell  com- 
prises the  entire  path  traversed  by  the  current,  including 
the  electrodes  and  the  liquid  in  the  cell  as  well  as  the  ex- 
ternal conductor.      Closing  the  circuit  means  joining  the 
two  electrodes  by  a  conductor ;  breaking  or  opening  the  cir- 
cuit is  disconnecting  them.     The  flow  of  current  in  the 
external  circuit  is  from  the  positive  electrode  (copper)  to 
the  negative  (zinc),  and  in  the  internal  part  of  the  circuit 
from  the  negative  electrode  to  the  positive  (Fig.  331). 

410.  Electrochemical  Actions  in  a  Voltaic  Cell.  —  The  mod- 
ern theory  of  dissociation  furnishes  an  explanation  of  the 
manner  in  which  an  electric  current  is  conducted  through 
a  liquid.     It  is  briefly  as  follows :  When  a  chemical  com- 
pound such  as  sulphuric  acid  (H2SO4),*  for  example,  is  dis- 
solved in  water,  some  of  the  molecules  at  least  split  into 

two  parts  (H^and  SO4,  for  example),  one  part  having 
a  positive  electrical  charge  and  the  other  a  negative  one. 

The  two  parts  of  the  dissociated  substance  with  their 
electrical  charges  are  called  ions  (from  a  Greek  word  mean- 
ing to  go).  An  electrolyte  is  a  compound  capable  of  such 
dissociation  into  ions.  It  conducts  electricity  only  by 
means  of  the  migration  of  the  ions  resulting  from  the  split- 
ting in  two  of  the  molecules.  The  separated  ions  convey 
their  charges  with  a  slow  and  measurable  velocity  through 

*  Each  molecule  of  sulphuric  acid  is  composed  of  two  atoms  of  hydrogen 
(H2),  one  of  sulphur  (S),  and  four  of  Oxygen  (O4). 


VOLTAIC  CELLS 


319 


Cu 


Zn 


(D<D(D 


the  liquid.  Electropositive  ions,  such  as  zinc  and  hydro- 
gen, carry  positive  charges  in  one  direction,  electronegative 

ions,  such  as  "sulphion"  (SO4),  carry  negative  charges 
in  the  opposite  direction,  and  the  sum  of  the  two  kinds  of 
charges  carried  through  the  liquid  per  second  is  the  meas- 
ure of  the  current. 

Fig.  333  represents  a  section  of  a  voltaic  cell  with  the 
electropositive  and  electronegative  ions.  When  the  cir- 
cuit is  closed  and  a  current  ~_  '* 

_^~ — •  ~~~^ 

flows,  zinc  from  the  zinc  plate 
enters  the  solution  as  electro- 

Vf  ^^x*" 

positive  ions  (Zn),  while  the 
positive  hydrogen  ions  migrate 
toward  the  copper  plate  or 
cathode,  and  the  sulphions  to- 
ward the  zinc  plate.  -  Tliu  QO4 " 
ions  carry  negative  charges  to  Fig-  333 

the  zinc  plate,  so  that  it  becomes  charged  negatively,  while 

the  44-.,  lose  carry  positive  charges  to  the  copper  plate  and  it 
becomes  charged  positively.  Zinc  from  the  zinc  plate  thus 
goes  into  solution  as  zinc  sulphate  (ZnSO4),  and  hydrogen 
when  it  has  given  up  its  positive  charge  is  set  free  as 
gaseous  hydrogen  on  the  copper  plate.  Some  prefer  to 
say  that  when  the  zinc  ions  with  their  positive  charge 
leave  the  zinc  plate,  the  equivalent  negative  is  left  behind 
to  charge  the  zinc  electrode.  The  zinc  ions  unite  with  the 
sulphions  to  form  neutral  zinc  sulphate.  Thus,  while  the 
zinc  ions  are  electropositive  and  carry  positive  charges, 
the  zinc  plate  is  charged  negatively. 

411.  Electromotive  Force.  —  Imagine  a  rotary  pump 
which  produces  a  difference  of  pressure  between  its  inlet 


320  ELECTRIC  CURRENTS 

and  its  outlet.  Such  a  pump  may  cause  water  to  circulate 
through  a  system  of  horizontal  pipes  against  friction.  In 
any  portion  of  the  pipe  system  the  force  producing  the 
flow  is  the  difference  of  water  pressure  between  the  ends 
of  that  portion.  But  the  force  is  all  applied  at  the  pump, 
and  this  produces  a  pressure  throughout  the  whole  circuit. 
A  voltaic  cell  is  an  electric  generator  analogous  to  such  a 
pump. 

A  voltaic  cell  generates  electric  pressure^called  electro- 
motive force.  It  does  not  generate  electricity  any  more 
than  the  pump  generates  water,  but  it  sup- 
plies  the  electric  pressure  to  set  electricity 
flowing.  This  electromotive  force  (E.M.F.) 
is  numerically  equal  to  the  work  which 
must  be  done  to  transport  a  unit  quantity 
of  electricity  around  the  external  circuit 
from  A  to  B,  through  the  zinc  plate  to  Z, 
from  Z  through  the  liquid  to  (7,  and  thence 


Fig.  334  back  to  A  (Fig.  334).  Work  is  done 
in  this  transfer,  because  all  conductors  offer  resistance  to 
the  passage  of  a  current.  The  energy  thus  expended  goes 
to  heat  the  conductor. 

412.  Difference  of  Potential.  —  The  difference  of  potential 
between  two  points,  A  and  B,  on  the  external  conducting 
circuit  is  the  work  done  in  carrying  a  unit  quantity  of 
electricity  from  the  one  point  to  the  other.  The  difference 
of  potential  between  the  electrodes  of  a  voltaic  cell  when 
the  circuit  is  closed  is  less  than  the  E.M.F.  of  the  cell  by 
the  work  done  in  transferring  unit  quantity  of  electricity 
through  the  electrolyte.  If  E  denotes  this  potential 
difference  and  Q  the  quantity  conveyed,  then  the  whole 
work  done  is  the  product  EQ.  But  the  quantity  con- 


VOLTAIC  CELLS  321 

veyed  by  a  conductor  per  second  is  called  the  strength  of 
current,  I.  The  energy  transformed  in  a  conductor,  there- 
fore, when  current  I  flows  through  it,  under  an  electric 
pressure  or  potential  difference  of  E  units  between  its 
ends,  is  ^Z"ergs  per  second. 

413.  Detection  of  Current.  —  Solder  a  copper  wire  to  each  of 
the  strips  of  a  voltaic  cell,  and  connect  the  wires  with  some  form  of 
key  to  close  the  circuit.      Stretch    a   portion   of  the  wire  over   a 
mounted      magnetic 

needle  (Fig.  335), 
holding  it  parallel 
to  it  and  as  near 
as  possible  without 
touching.  Now  close 
the  circuit;  the 
needle  is  deflected, 

and  comes  to  rest  at 

.,,     .,  Fig.  335 

an    angle    with    the 

wire.  Next  form  a  rectangular  loop  of  the  wire,  and  place  the  needle 
within  it.  A  greater  deflection  is  now  obtained.  If  a  loop  of 
several  turns  is  formed,  the  deflection  is  still  greater. 

A  magnetic  needle  employed  in  this  way  becomes  a 
galvanoscope,  a  detector  of  electric  currents.  This  experi- 
ment, first  performed  by  Oersted  in  1819,  shows  that  the 
region  around  the  wire  has  magnetic  properties  during  the 
flow  of  electricity  through  the  wire.  In  other  words,  it  is 
a  magnetic  field  (§  362). 

414.  Relation  between  the   Direction  of  the  Current  and 
the  Direction  of   Deflection.  —  Making  use  of  the  apparatus  of 
§  410,  compare  the  direction  of  the  current  through  the  wire  with 
that  in  which  the  north  pole  of  the  needle  turns.     Cause  the  current 
to  pass  in  the  reverse  direction  over  the  needle ;   the  deflection  is 
reversed.     Now  hold  the  wire  below  the  needle,  and  the  direction 
of  deflection  is  again  reversed  as  compared  with  the  deflection  when 
the  wire  is  held  above  the  magnetic  needle. 


322  ELECTRIC  CURRENTS 

The  direction  of  the  deflection  may  always  be  predicted 
by  the  following  rule  :   Stretch  out  the  right  hand  along  the 

wire,  with  the  palm  turned 
toward  the  magnetic  needle, 
and  with  the  current  flowing 
in  the  direction  of  the  ex- 
tended fingers.  The  out- 
stretched thumb  will  then 
Fi  336  point  in  the  direction  in 

which  the  north  pole  of  the 

needle  is  deflected  (Fig.  336).  By  the  converse  of  this 
rule,  the  direction  of  the  current  may  be  inferred  from  the 
direction  in  which  the  needle  is  deflected. 


415.  Local  Action.  —  Place  a  strip  of  commercial  zinc  in  dilute 
sulphuric  acid.'  Hydrogen  is  liberated  during  the  chemical  action, 
and  after  a  few  minutes  the  zinc  becomes  black  from  particles  of 
carbon  exposed  to  view  by  dissolving  away  the  surface.  If  the 
experiment  is  repeated  with  zinc  amalgamated  with  mercury,  that 
is,  by  coating  it  with  an  alloy  of  mercury  and  zinc,  there  will  be 
little  or  no  chemical  action.  A  strip  of  chemically  pure  zinc  acts 
much  like  one  amalgamated  with  mercury. 

Thus  we  see  that  the  amalgamation  of  commercial  zinc 
with  mercury  changes  its  properties.  If  in  the  experiment 
with  the  simple  voltaic  cell,  a  galvanoscope  is  inserted  in 
the  circuit  both  before  the  zinc  has  been  amalgamated  and 
afterward,  it  will  be  found  that  a  larger  deflection  will  be 
obtained  in  the  second  case. 

In  a  voltaic  cell  the  chemical  action  which  contributes 
nothing  to  the  current  flowing  through  the  circuit  is 
known  as  local  action.  It  is  probably  due  to  the  presence 
of  carbon,  iron,  etc.,  in  the  zinc ;  these  with  the  zinc  form 
miniature  voltaic  cells,  the  currents  flowing  around  in  short 


Hans  Christian  Oersted  (1777-1851)  was  born  at  Rudkjo- 
bing,  Denmark,  and  received  his  education  at  the  University  of 
Copenhagen,  afterward  becoming  professor  in  the  University  and 
polytechnic  schools  of  that  city.  It  was  while  holding  this  posi- 
tion that  he'  discovered  the  action  of  the  electric  current  on  the 
magnetic  needle,  thus  establishing  the  connection  between  elec- 
tricity and  magnetism  which  had  long  been  sought  by  scientists. 
He  also  discovered  that  this  magnetic  action  of  the  electric  cur- 
rent takes  place  freely  through  a  great  many  substances.  Oersted 
wrote  extensively  for  newspapers  and  magazines  in  an  endeavor 
to  make  science  popular. 


VOLTAIC  CELLS 


323 


circuits  from  the  zinc  through  the  liquid  to  the  foreign 
particles  and  back  to  the  zinc  again. 

This  local  action  is  prevented  by  amalgamating  the  zinc. 
The  amalgam  brings  pure  zinc  to  the  surface,  covers  the 
foreign  particles,  and  above  all  forms  a  smooth  surface,  so 
that  a  film  of  hydrogen  clings  to  it  and  protects  it  from 
chemical  action  save  when  the  circuit  is  closed. 

416.  Polarization.  —  Connect  the  poles  of  a  voltaic  cell  to  a  gal- 
vanoscope  and  note  the  deflection.  Let  the  cell  remain  in  circuit  with 
the  galvanoscope  for  some  time ;  the  deflection  will  gradually  become 
less  and  less.  Now  stir  up  the  liquid  vigor- 
ously with  a  glass  rod,  inserting  the  rod  be- 
tween the  plates  and  brushing  off  the  adhering 
gas  bubbles ;  the  deflection  will  increase  nearly 
to  its  first  value. 

Fasten  two  strips  of  zinc  and  two  of  copper 
to  a  square  board  and  immerse  them  in  dilute 
sulphuric  acid  (Fig.  337).  Join  one  zinc  and 
one  copper  strip  with  a  short  wire  for  a  few 
minutes.  Then  disconnect  and  join  the  two 
coppers  to  a  galvanoscope.  The  direction  of 
the  deflection  will  be  the  same  as  if  zinc  were 
used  in  place  of  the  copper  strip  coated  with 
hydrogen.  The  hydrogen-coated  copper  acts 
like  zinc  and  tends  to  produce  a  current  through  the  electrolyte  from 
it  to  the  copper  free  from  hydrogen. 

The  diminution  in  the  intensity  of  the  current  is  due  to 
several  causes,  but  the  chief  one  is  the  film  of  hydrogen 
which  gathers  on  the  copper  plate,  causing  what  is  known 
as  the  polarization  of  the  cell.  The  hydrogen  on  the  posi- 
tive plate  not  only  introduces  more  resistance  to  the  flow  of 
the  current,  but  it  diminishes  the  electromotive  force  to 
which  this  flow  is  due.  The  presence  of  hydrogen  on 
the  copper  plate  sets  up  an  inverse  E.M.F.,  which  reduces 
the  flow. 


324 


ELECTRIC   CURRENTS 


417.  Remedies  for  Polarization.  —  Place  enough  pure  mercury 
in  a  quart  jar  to  cover  the  bottom,  and  hang  above  it  apiece  of  sheet 
zinc.  Fill  the  jar  with  a  nearly  saturated  solution  of  salt  water,  and 
place  in  the  mercury  the  exposed  end  of  a  copper  wire  insulated  with 
gutta-percha,  the  mercury  forming  the  positive  electrode  of  the  battery. 

If  now  the  circuit  is  closed  through  a  telegraph  sounder  (§  494)  of 
ten  or  fifteen  ohms  resistance,  the  armature  will  at  first  be  attracted 
strongly ;  but  in  the  course  of  a  few  minutes  it  will  be  released  and 
will  be  drawn  back  by  the  spring.  Polarization  has  then  set  in  to 
the  extent  that  the  current  is  insufficient  to  operate  the  instrument. 

Next  take  a  small  piece  of  mercuric  chloride  (HgCl-2)  no  larger 
than  the  head  of  a  pin,  and  drop  it  in  on  the  surface  of  the  mercury. 
The  armature  of  the  sounder  will  instantly  be  drawn  down,  showing 
that  the  current"  has  recovered  its  normal  value.  The  hydrogen  has 
been  removed  by  the  chlorine  of  the  mercuric  chloride.  In  a  few 
minutes  the  chlorine  will  be  exhausted,  and  polarization  will  again 
set  in.  A  little  more  of  the  chloride  will  again  restore  the  activity  of 
the  cell.  (This  experiment  was  devised  several  years  ago  by  Mr.  D.  H. 
Fitch.) 

This  illustrates  a  chemical  method  of  reducing  polariza- 
tion. The  hydrogen  ions  are  replaced  by  others,  such  as 
copper  or  mercury,  which  do  not 
produce  polarization  when  they  are 
deposited  on  the  positive  electrode  ; 
or  else  the  positive  electrode  is  sur- 
rounded with  a  chemical  which  fur- 
nishes oxygen  or  chlorine  to  unite 
with  the  hydrogen  before  it  reaches 
the  electrode.  In  both  cases  the 
electrode  is  kept  nearly  free  from 
hydrogen. 

418.  The  Daniell  Cell.  —  Pie  Daniell 
Cell  in  its  most  common  form  (Fig. 

338)  consists  of  a  glass  jar  containing  a  saturated  solution 
of  copper  sulphate  (CuSO4),  and  in  it  a  cylinder  0  of 


VOLTAIC  CELLS 


325 


copper,  which  is  cleft  down  one  side.  Within  the  copper 
cylinder  is  a  porous  cup  of  unglazed  earthenware  con- 
taining a  dilute  solution  of  zinc  sulphate  (ZnSO4).  In 
the  porous  cup  also  is  the  zinc  prism  Z.  The  copper 
sulphate  must  not  be  allowed  to  come  in  contact  with  the 
zinc  electrode.  The  porous  cup  allows  the  ions  to  pass 
through  its  pores,  but  it  prevents  the  rapid  admixture 
of  the  two  sulphates. 

Both  electrolytes  undergo  partial  dissociation  into  ions; 
and  when  the  circuit  is  closed,  the  zinc  and  the  copper 
ions  both  travel  toward  the  copper  electrode.  The  zinc 
ions  do  not  reach  the  copper,  because  zinc  in  copper  sul- 
phate replaces  copper,  forming  zinc  sulphate.  The  result 
is  the  formation  of  zinc  sulphate  at  the  zinc  electrode  and 
the  deposition  of  metallic  copper  on  the  copper  electrode. 
Polarization  is  quite  completely  obviated;  and,  so  long  as 
the  circuit  is  kept  closed,  the  mixing  of  the  electrolytes 
by  diffusion  is  slight.  This  cell  must  not  be  left  on  open 
circuit  because  the  copper  sulphate  then  diffuses  until 
it  reaches  the  zinc  and  causes  a 
black  deposit  of  copper  oxide  on  it. 

419.  The  Gravity  Cell.  —  This  cell 
(Fig.  339)  is  a  modified  Daniell. 
The  porous  cup  is  omitted,  and  the 
partial  separation  of  the  liquids  is 
secured  by  difference  in  density. 
The  copper  electrode  0  is  placed  at 
the  bottom  in  saturated  copper  sul- 
phate B,  while  the  zinc  Z  is  sus- 
pended near  the  top  in  a  weak  solution  of  zinc  sulphate  A, 
floating  on  top  of  the  copper  sulphate.  The  zinc  should 
never  be  placed  in  the  solution  of  copper  sulphate.  The 


•• 


326 


ELECTRIC  CURRENTS 


saturated  copper  sulphate  is  more  dense  than  the  dilute 
zinc  salt,  and  so  remains  at  the  bottom,  except  as  it 
slowly  diffuses  upward. 

420.  The  Leclanche  Cell  consists  of  a  glass  vessel  contain- 
ing a  saturated  solution  of  ammonium  chloride  (sal  am- 
moniac) in  which  stands   a   zinc  rod 
and  a  porous  cup  (Fig.  340).     In  this 
porous  cup  is   a  bar   of   carbon   very 
tightly  packed  in  a  mixture  of  man- 
ganese dioxide  and  graphite,  or  granu- 
lated carbon. 

The  zinc  is  acted  on  by  the  chlorine 
of  the  ammonium  chloride,  liberating 
ammonia  and  hydrogen.  The  ammonia 
in  part  dissolves  in  the  liquid,  and  in 
part  escapes  into  the  air.  The  hydro- 
gen is  slowly  oxidized  by  the  man- 
The  cell  is  not 

adapted  to  continuous  use,  as  the 

hydrogen  is  liberated  at  the  posi- 

tive  electrode  faster  than  the  oxida- 

tion  goes  on,  and  hence  the  cell 

polarizes.  If,  however,  it  is  allowed 

to  rest,  it  recovers  from  polariza- 
tion. The  Leclanche  cell  is  suitable 

for  ringing  electric  bells. 

421.  The  Dry  Cell.—  The  "dry  " 
cell  is  merely  a  modified  Leclanche 
cell  adapted  for  use  in  situations 
where    cells   with    a    liquid    elec- 
trolyte   could    not   be   employed. 

The  electrodes  are  zinc  and  carbon,  Fig.  341 


Fig.  340 
ganese  dioxide. 


-E 


ELECTR  OL  TSIS  327 

the  former  being  the  cylindrical  vessel  containing  the 
other  parts  of  the  cell.  The  zinc  Z  (Fig.  341)  is  con- 
tained in  a  cardboard  case  and  has  a  thin  insulating 
layer  E  at  the  bottom.  The  carbon  plate  is  flat ;  it  is 
shown  edgeways  in  the  figure.  Between  the  two  plates 
is  a  moist  paste  composed  of  zinc  oxide,  sal  ammoniac, 
zinc  chloride,  plaster  of  Paris,  and  water.  About  twenty- 
five  per  cent  of  this  paste  is  water.  The  oxide  of  zinc 
makes  the  composition  porous,  and  so  aids  in  the  escape 
of  gases  and  helps  to  prevent  polarization.  Graphite  and 
manganese  dioxide  are  also  sometimes  added  to  the  paste. 
The  cell  is  sealed  with  a  bituminous  compound  H,  through 
which  is  a  vent  V  for  the  escape  of  gases. 

71'x   'vj^vd-  V  f  VUU^^OiM,    Urtav^    a 

-CUL     yvULvA.4.:  c*  •' 

II.   ELECTROLYSIS 

422.  Phenomena  of  Electrolysis.  —  Thrust  platinum  wires 
through  the  corks  closing  the  ends  of  a  V-tube  (Fig.  342).  Fill  the 
tube  nearly  full  with  a  solution  of  sodium  sulphate  colored  with  blue 
litmus.  Pass  through  it  a  current 
for  a  few  minutes.  The  liquid  around 
the  anode  where  the  current  enters 
will  turn  red,  showing  the  formation 
of  an  acid;  the  liquid  around  the 
cathode  where  the  current  leaves  := 
the  cell  will  turn  a  darker  blue, 
showing  the  presence  of  an  alkali.  Fig.  342 

The  electric  current  in  its  passage  through  a  liquid 
decomposes  it.  This  process  of  decomposing  a  liquid  by 
an  electric  current  Faraday  named  electrolysis;  the  liquid 
decomposed  he  called  the  electrolyte;  the  parts  of  the 
separated  electrolyte,  ions.  The  current  enters  the  elec- 
trolyte by  the  anode  and  leaves  it  by  the  cathode.  The 
experiment  illustrates  certain  polarity  indicators,  which 


328  ELECTRIC  CURRENTS 

are  made  of  a  short  tube  filled  with  a  suitable  electrolyte 
and  contain  electrodes  terminating  in  binding  posts  at  the 
two  ends  (Fig.  343).  When  this  is  placed  in  circuit,  the 
liquid  turns  red  at  one  pole. 

423.   Electrolysis  of  Copper  Sulphate.  —  Fill  the  V-tube  of  the 

last  experiment  about  two- 
thirds  full  of  a  solution  of 
copper  sulphate.  After  the 
circuit  has  been  closed  a  few 
PJ~  343  minutes,  the  cathode  will  be 

covered  with  a  deposit  of  cop- 
per, and  bubbles  of  gas  will  rise  from  the  anode.  These  bubbles  are 
oxygen. 

When  copper  sulphate  is  dissolved  in  water  it  is  dis- 
sociated to  some  extent.     If,  therefore,  electric  pressure 

is  applied  to  the   solution    through   the    electrodes,    the 

+ 
electropositive  ions  (Cu)  are  set  moving  from  higher  to 

lower  potential,  while  the  electronegative  ions  (SO4)  carry 

+ 
their  negative  charges  in  the  opposite  direction.     The  Cu 

ions  are  therefore  driven  against  the  cathode,  and,  giving 
up  their  charges,  become  metallic  copper.  The  sulphions 

(SO4)  go  to  the  anode;  and,  giving  up  their  charges, 
they  take  hydrogen  from  the  water  present,  forming  sul- 
phuric acid  (H2SO4)  and  setting  free  oxygen,  which 
comes  off  as  bubbles  of  gas.  If  the  anode  were  copper 
instead  of  platinum,  the  sulphion  would  unite  with  it, 
forming  copper  sulphate,  and  copper  would  then  be  re- 
moved from  the  anode  as  fast  as  it  is  deposited  on  the 
cathode.  The  result  of  the  passage  of  a  current  would 
then  be  the  transfer  of  copper  from  the  anode  to  the 
cathode.  This  is  what  takes  place  in  the  electrolytic 
refining  of  copper. 

Thus  the  passage   of  an   electric    current  through   an 


ELECTROLYSIS 


329 


electrolyte  is  accomplished  in  the  same  way,  whether  it  is 
in  a  voltaic  cell  or  in  an  electrolytic  cell. 

424.  Electrolysis  of  Water.  —  Water  appears  to  have  been 
the  first  substance  decomposed  by  an  electric  current.     Pure 
water  does  not  conduct  an  appreciable 

current  of  electricity,  but  if  it  is  acid- 
ulated with  a  small  quantity  of  sulphuric 
acid,  electrolysis  takes  place. 

In  HofmannV  apparatus  (Fig.  344)  the 
acidulated  water  is  poured  into  the  bulb  at  the 
top,  and  the  air  escapes  by  the  glass  taps  until 
the  tubes  are  filled.  The  electrodes  at  the 
bottom  in  the  liquid  are  platinum  foil.  If  a 
current  is  sent  through  the  liquid,  bubbles  of 
gas  will  be  liberated  on  the  pieces  of  platinum 
foil.  The  gases  collecting  in  the  tubes  may  be 
examined  by  letting  them  escape  through  the 
taps.  Oxygen  will  be  found  at  the  anode 
and  hydrogen  at  the  cathode;  the  volume  of 
the  hydrogen  will  be  nearly  twice  that  of 
the  oxygen. 

425.  Laws  of  Electrolysis.  —  The  fol-  Fig<  344 
lowing  laws  of  electrolysis  were  established  by  Faraday : 

I.  The  mass  of  an  electrolyte  decomposed  by  an  electric 
current  is  proportional  to  the  quantity  of  electricity  con- 
veyed through  it. 

The  mass  of  an  ion  liberated  in  one  second  is,  therefore, 
proportional  to  the  strength  of  current. 

II.  When  the  same  quantity  of  electricity  is  conveyed 
through  different  electrolytes,  the  masses  of  the  different 
ions  set  free  at  the  electrodes  are  proportional  to  their 
chemical  equivalents. 

By    u  chemical    equivalents "    are    meant    the   relative 


330  ELECTRIC  CURRENTS 

quantities  of  the  ions  which  are  chemically  equivalent  to 
one  another,  or  take  part  in  equivalent  chemical  reactions. 
Thus,  32.5  gm.  of  zinc  or  31.7  gm.  of  copper  take  the 
place  of  one  gm.  of  hydrogen  in  sulphuric  acid  (H2SO4)  to 
form  zinc  sulphate  (ZnSO4)  or  copper  sulphate  (CuSO4), 
respectively. 

The  first  law  of  electrolysis  affords  a  valuable  means  of 
comparing  the  strength  of  two  electric  currents  by  deter- 
mining the  relative  masses  of  any  ion,  such  as  silver  or 
copper,  deposited  by  the  two  currents  in  succession  in  the 
same  time  (§  433). 

426.  Electroplating  consists  in   covering  bodies  with  a 
coating  of  any  metal  by  means  of  the  electric   current. 
The  process  may  be  summarized  as  follows :     Thoroughly 
clean  the  surface  to  remove  all  fatty  matter.      Attach  the 
article  to  the  negative  electrode  of  a  battery,  and  suspend 
it  in  a  solution  of  some  chemical  salt  of  the  metal  to  be 
deposited.     If  silver,  cyanide  of  silver  dissolved  in  cyanide 
of  potassium  is  used ;  if  copper,  sulphate  of  copper.     To 
maintain  the  strength  of  the  solution  a  piece  of  the  metal 
of  the  kind  to  be  deposited  is  attached  to  the  positive 
electrode  of  the  battery.     The  action  is  similar  to  that 
heretofore  given.     Articles  of  iron,  steel,  zinc,  tin,  and  lead 
cannot  be  silvered  or  gilded  unless  first  covered  with  a  thin 
coating  of  copper. 

All  silver  plating,  nickeling,  gold  plating,  and  so  on,  is 
done  by  this  process. 

427.  Electrotyping   consists   in  copying   medals,   wood- 
cuts, type,  and  the  like  in  metal,  usually  copper,  by  means 
of  the  electric  current.     A  mold  of  the  object  is  taken  in 
wax  or  plaster  of   Paris.     This   is   evenly  covered  with 
powdered  graphite  to  make  the  surface  a  conductor,  and 


ELECTROLYSIS 


331 


treated  very  much  as  an  object  to  be  plated.  When  the 
deposit  has  become  sufficiently  thick  it  is  removed  from 
the  mold  and  backed  or  filled  with  type-metal. 

Nearly  all  books  nowadays  are  printed  from  electrotype 
plates,  and  not  as  formerly  from  movable  types. 

428.  The  Storage  Cell.  —  Attach  two  lead  plates,  to  which  are 

soldered  copper  wires,  to  the  opposite  sides  of  a  block  of  dry  wood, 

and  immerse  them   in    dilute   sulphuric 

acid,  one  part  acid  to  five  of  water  (Fig. 

345).      Connect  this  cell  to   a   suitable 

battery  B  by  means  of  key  K\ ;    also  to 

an  ordinary  electric  house  bell  H  through 

a  key  K2  (Fig.  346).     A    galvanoscope 

G    may    be    included    in    the  circuit  to 

show  the  direction  of  the  current.     Pass 

a  current  through  the  lead  cell  for  a  few 

minutes  by  closing  the  key  Kv  Hydro- 
gen bubbles  will  be  disengaged  from  the 

cathode,  while   the  anode  will  begin  to 

turn  dark  blbwn.         Next  open  the  key 

Kit    thus   disconnecting  the   battery  B, 

and  close  key  K2.     The    bell   will   ring  and  the  galvanoscope  will 

indicate  a  discharge  current  in  the  opposite  direction  to  the  first  or 

charging  current.  The  bell  will 
soon  cease  ringing,  and  the  charg- 
ing may  be  repeated  by  again 
closing  key  K^  while  K%  is  open. 

The  lead  plates  in  an  elec- 
trolyte of  sulphuric  acid  il- 
lustrate a  simple  lead  storage 
cell.  The  electrolysis  of  the 


Fig.  345 


Fig.  346 


sulphuric  acid  liberates  oxygen  at  the  anode,  which  com- 
bines with  the  lead  electrode  to  form  a  chocolate-colored 
deposit  of  lead  peroxide  (FbO2).  Hydrogen  accumulates 
on  the  cathode.  When  the  charging  battery  is  discon- 
nected and  the  lead  plates  are  joined  by  a  conductor,  a 


332  ELECTRIC  CURRENTS 

current  flows  in  the  external  circuit  from  the  chocolate- 
colored  plate,  which  is  called  the  positive  electrode,  to  the 
other  one,  called  the  negative;  the  lead  peroxide  is  reduced 
to  spongy  lead  on  the  positive  plate,  while  some  lead  sul- 
phate is  formed  on  the  negative.  During  subsequent 
charging  this  lead  sulphate  is  reduced  by  the  hydrogen  to 
spongy  lead.  Note  that  the  charging  current  passes 
||  ||  through  the  storage  cell  in  the  oppo- 

site direction  to  the  discharge  current 
4^"ifP-~-  *ljr^      furnished  by  the  cell  itself. 

The  storage  battery  stores  energy 
and  not  electricity.  The  energy  of 
the  charging  current  is  converted 
into  the  potential  energy  of  chemical 
separation  in  the  storage  cell.  When 
the  circuit  of  the  charged  secondary 
cell  is  closed,  the  potential  chemical 
energy  is  reconverted  into  the  energy 
of  an  electric  current  in  precisely  the 
same  way  as  in  a  primary  cell. 

Fig.  347  shows  a  complete  storage  cell  containing  one 
positive  and  two  negative  plates. 

III.     OHM'S  LAW  AND  ITS  APPLICATIONS 

429.  Resistance. — Every  conductor  presents  some  ob- 
struction to  the  passage  of  electricity.  This  obstruction 
is  called  its  electrical  resistance.  The  greater  the  con- 
ductance of  a  conductor  the  less  its  resistance,  the  one 
decreasing  in  the  same  ratio  as  the  other  increases. 
Resistance  is  the  reciprocal  of  conductance.  If  R  is  the 
resistance  of  a  conductor  and  C  its  conductance,  then 


OHM'S  LAW  AND  ITS  APPLICATIONS  333 

430.  Unit  of  Resistance.  —  The  practical  unit  of  resist- 
ance is  the  ohm.     It  is  represented  by  the  resistance  offered 
to  an  unvarying  current  by  a  thread  of  mercury  at  the  tem- 
perature of  melting  ice,  one  square  millimeter  in  cross  sec- 
tional area,  and  of  a  length  of  106.3  cm.      Mercury  has 
been  chosen  because  it  can  be  obtained  in  great  purity, 
and  the  cross  section  of  the  glass  tube  containing  it  can 
be  measured  with  the  highest  degree  of  accuracy. 

431.  Laws  of  Resistance.  —  1.    The  resistance  of  a  con- 
ductor is  proportional  to  its  length.     For  example,  if  39  ft. 
of  No.  24  copper  wire  (B.  &  S.  gauge)  have  a  resistance 
of  1  ohm,  then  78  ft.  of  the  same  wire  will  have  a  resist- 
ance of  2  ohms. 

2.  The  resistance  of  a  conductor  is  inversely  propor- 
tional to  its  cross  sectional  area.     In  the  case  of  round 
wire  the'  resistance  is  proportional  to  the  square  of  the 
diameter.     For  example,  No.  24  copper  wire  has  twice  the 
diameter  of  No.  30.     Then  39  ft.  of  No.  24  has  a  resist- 
ance of  1  ohm,  and  9.75  ft.  of  No.  30  (one  fourth  of  39) 
also  has  a  resistance  of  1  ohm,  both  at  22°  C. 

3.  The  resistance  of  a  conductor  of  given  length  and 
cross  section  depends  upon  the  material  of  which  it  is 
made,  and  is  affected  by  anything  which  modifies  its 
molecular  condition.     For  example,  the  resistance  of  2.2 
ft.  of  No.  24  German  silver  wire  is  1  ohm,  while  it  takes 
39  ft.  of  copper  wire  of  the  same  diameter  to  give  the 
same  resistance.     Heat  affects  the  molecular  condition  of 
a  conductor,  and  consequently  affects  its  resistance.    Metal 
conductors  have  their  resistance  increased  by  a  rise   of 
temperature,  but  all  are  not  affected  to  the  same  extent. 
The  resistance  of  a  copper  conductor  increases  much  more 
for  a  given  rise  of  temperature  than  one  of  German  silver ; 


334  ELECTRIC  CURRENTS 

other  alloys,  such  as  manganin,  show  very  little  change 
with  temperature.  The  resistance  of  carbon  and  of  elec- 
trolytes decreases  with  a  rise  of  temperature.  The  resist- 
ance of  a  hot  carbon  filament  in  an  incandescent  lamp  is 
only  about  half  as  great  as  when  it  is  cold. 

432.  Formula  for  Resistance.  —  The  above  laws  are  con- 
veniently expressed  in  the  following  formula  for  the  re- 
sistance of  a  wire  : 


in  which  k  is  a  constant  depending  on  the  material,  I  the 
length  of  the  wire  in  feet,  and  C.M.  denotes  "  circular 
mils."  A  "mil"  is  a  thousandth  of  an  inch,  and  circular 
mils  are  the  square  of  the  mils;  that  is,  the  square  of  the 
diameter  of  the  wire  in  thousandths  of  an  inch.  For  ex- 
ample, if  the  diameter  of  a  wire  is  0.020  in.,  then  in  mils 
it  is  20,  and  the  circular  mils  (C.M.)  will  be  the  square  of 
20  or  400.  Now  if  the  length  of  a  wire  conductor  is  ex- 
pressed in  feet,  and  its  cross  section  in  circular  mils,  then 
it  is  easy  to  give  to  k  for  each  kind  of  conductor  such  a 
value  that  R  in  the  above  formula  will  be  in  ohms. 

The  following  are  the  values  of  k  in  ohms  for  several 
metals,  at  20°  C.: 

Silver         9.53         Iron  61.3      German  silver  181.3 

Copper     10.19         Platinum    70.5      Mercury  574 

433.  Strength  of  Current.  —  The  strength  or  intensity  of 
a  current  is  measured  by  the  magnitude  of  the  effects  pro- 
duced by  it.  Any  such  effect  may  be  made  the  basis  of  a 
system  of  measurement.  The  quantity  of  an  ion  deposited 
in  a  second  is  a  convenient  one  to  use  in  defining  unit 
strength  of  current.  The  unit  of  current  strength  is  the 


OHM'S  LAW  AND  ITS  APPLICATIONS 


335 


ampere.  It  is  defined  as  the  current  which  will  deposit 
by  electrolysis,  under  suitable  conditions,  0.001118  gm.  of 
silver  per  second.  The  ampere  deposits  4.025  gm.  of 
silver  in  one  hour.  A  milliampere  is  a  thousandth  of  an 
ampere.  It  is  to  be  noted  that  the  electrolytic  method 
measures  only  the  quantity  of  electricity  passing  through 
the  decomposing  cell,  called  a  voltameter,  in  the  given  time. 

434.  Electromotive  Force  is  the  c.ause  of  an  electric  flow. 
It  is  often  called  electric  pressure  from  its  superficial  anal- 
ogy to  water  pressure.  The  unit  of  electromotive  force 
(E.M.F.)  is  the  volt.  A  volt  is  the  E.M.F.  which  will 
cause  a  current  of  one  ampere  to  flow  through  a  resistance  of 
one  ohm.  The  E.M.F.  of  a  voltaic  cell  depends  upon  the 
materials  employed,(and  is  entirely  independent  of  the  size 
and  shape  of  the  platesA  The  E.M.F.  of  a  Daniell  cell 
and  of  a  gravity  cell  is  about  IJ^joJts ;  of  a  Leclanche 
and  of  a  dry  cell,  1.5  volts;  of  a  lead  storage  cell,  2  volts. 

The  practical  international  standard  of  electromotive 
force  is  the  Weston  Normal  Cell.  The  electrodes  are 
cadmium  amalgam 
for  the"  negative 
and  mercury  for 
the  positive.  The 
electrolyte  is  a  sat- 
urated solution  of 
cadmium  sulphate, 
and  the  depolarizer 
is  mercurous  sul- 
phate (Fig.  348). 


Cadmium  sulphate  _ 
crystals  and  solution 


Cadmium  A 


Mercurous  sulphate 
paste 

Platinum  wire, 
Mercury 


Fig.  348 

The  E.M.F.  of  the  Weston  cell  in 
volts  is  given  by  the  following  equation,  the  temperature 
t  being  in  centigrade  degrees: 

E  =  1.0183  -  0.00004  (t  -  20°).     (Equation  35) 


336  ELECTEIC  CURRENTS 

435.  Ohm's   Law. — The   definite  relation  existing   be- 
tween  strength   of   current,    resistance,    and    E.M.F.    is 
known  as  Ohm's  Law: 

The  strength  of  a  current  equals  the  electromotive 
force  divided  by  the  resistance;  then 

E.M.F.  (or  potential  difference}  in  volts 

current  in  amperes  =  —  — *-— - —        -^ — , 

resistance  in  ohms 

-pi 

or  in  symbols,  /=  — ,         .      .     .     .      (Equation  36) 

where  /is  the  current  in  amperes,  E  the  E.M.F.  in  volts, 
and  R  the  resistance  in  ohms.  Applied  to  a  battery,  if 
R  is  the  resistance  external  to  the  cell,  and  r  the  internal 
resistance  of  the  cell  itself,  then 

I=     % 
From  equation  (36),  E  =  IE  and  R  =  j. 

436.  Methods  of  Varying  Strength  of  Current.  —  It  is  evi- 
dent from  Ohm's  law  that  the    strength  of   the.  current 
furnished  b}^  an   electric  generator  may  be  increased  in 
two  ways  :  (1)  by  increasing  the  E.M.F. ;  (2)  by  reducing 
the  internal  resistance. 

The  E.M.F.  may  be  increased  by  joining  several  cells 
in  series,  and  the  internal  resistance  may  be  diminished  by 
connecting  them  in  parallel.  Enlarging  the  plates  of  a 
battery  or  bringing  them  closer  together  diminishes  the 
internal  resistance. 

437.  Connecting  in  Series.  —  To  connect  cells  in  series, 
join  the  positive  electrode  of  one  to  the  negative  electrode 
of  the  next,  and  so  on  until  all  are  connected.     The  elec- 


OHM'S  LAW  AND  ITS  APPLICATIONS 


337 


trodes  of  the  lattery  thus  connected  in  series  are  the  posi- 
tive electrode  of  the  last  one  in  the  series  and  the  negative 
electrode  of  the  first  one 
(Fig.  349).  Fig.  350  is  the 
conventional  sign  for  a  single 
cell;  Fig.  351  shows  four 
cells  in  series. 

When  n  similar  cells  are 
connected  in  series,  the 
E.  M.  F.  of  the  battery  is  n 
times  that  of  a  single  cell ; 
the  resistance  is  also  n  times 

the  resistance  of  one  cell.     Hence,  by  Ohm's  law  for  n 
cells  connected  in  series  the  current  is 

nE 


R  +  nr 

To  illustrate,  if  four  cells,  each  having  an  E.  M.  F.  of 
2  volts  and  an  internal  resistance  of  0.5  ohm,  are  joined 


V, 


4- 


Fig.  350 


Fig.  351 


in  series  with  an  external  resistance  of  10  ohms,  the  cur- 
rent will  be 

4x2 


1= 


10  +  4  xO.5 


=  0.67  ampere. 


438.    Connecting  in   Parallel.  —  When  all   the   positive 
terminals  are  connected   together   on   one  side  and  the 


338  ELECTRIC  CURRENTS 

negative  on  the  other,  the  cells  are  grouped  in  parallel 
(Fig.   352).     With  n  similar  cells  the  effect  of  such  a 

grouping    is    to    reduce    the 

internal  resistance  to  — th  that 
n 

of  a  single  cell.  It  is  equiv- 
alent to  increasing  the  area 
of  the  plates  n  times.  All 
the  cells  side  by  side  con- 
Fig-  352  tribute  equal  shares  to  the 

output  of   the  battery.     The    E.  M.  F.    of   the  group  is 

the  same  as  that  of  a  single  cell. 

Connection  in  parallel  is  used  chiefly  with  storage  cells,  not  for  the 
purpose  of  reducing  the  internal  resistance  of  the  battery,  but  for  the 
purpose  of  permitting  a  larger  current  to  be  drawn  from  it  with 
safety  to  the  cells.  The  ampere  capacity  of  a  storage  cell  depends  on 
the  area  of  the  plates.  If  twenty  amperes  may  be  drawn  from  a 
single  storage  cell,  then  from  two  such  cells  in  parallel  forty  amperes 
may  be  taken. 


IV.     HEATING  EFFECTS   OF  A  CURRENT 

439.  Electric  Energy  Converted  into  Heat.  —  Send  an  electric 
current  through  a  piece  of  fine  iron  wire.  The  wire  is  heated,  and  it 
may  be  fused  if  the  current  is  sufficiently  strong. 

The  conversion  of  electrical  energy  into  other  forms  is 
a  familar  fact.  In  the  storage  battery  the  energy  of  the 
charging  current  is  converted  into  the  energy  of  chemical 
separation  and  stored  as  the  potential  energy  of  the 
charged  cells.  In  this  experiment  the  energy  of  the  cur- 
rent is  transformed  into  heat  because  of  the  resistance 
which  the  wire  offers.  If  the  resistance  of  an  electric 
circuit  is  not  uniform,  the  most  heat  will  be  generated 
where  the  resistance  is  the  greatest. 


HEATING  EFFECTS   OF  A    CURRENT 


339 


440.  Joule's  Law.  —  Joule  demonstrated  experimentally 
that  the  number  of  units  of  heat  generated  in  a  conductor 
by  an  electric  current  is  proportional : 

a.  To  the  resistance  of  the  conductor. 

b.  To  the  square  of  the  strength  of  current. 

c.  To  the  length  of  time  the  current  flows.1 

441.  Applications  of  Electric  Heating.  —  Some  of  the  more 
important  applications  of  electric  heating  are  the  following: 

1.  Electric  Cautery.      A  thin  platinum  wire  heated  to  incandescence 
is  employed  in  surgery  instead  of  a  knife.     Platinum  is  very  infusible 
and  is  not  corrosive. 

2.  Safety   Fuses.     Advantage   is   taken  of  the   low  temperature 
of  fusion  of  some  alloys,  in  which  lead   is   a 

constituent,  for  making  safety  fuses  to  open 
a  circuit  automatically  whenever  the  current 
becomes  excessive. 

3.  Electric    Heating.     Electric    street   cars 
are  often  heated  by  a  current  through  suitable 
resistances.     Similar  devices  for  cooking  are 
now  articles  of  commerce.      Small  furnaces 
for  fusing,   vulcanizing,   and  enameling  are 
now  common  in  dentistry. 

Large  furnaces  are  employed  for  melting 
refractory  substances,  for  the  reduction  of  cer- 
tain ores,  and  for  chemical  operations  de- 
manding a  high  temperature. 

4.  Electric  Welding.     If  the  abutting  ends 
of    two    rods  or  bars  are   pressed  together, 
while   a   large  current  passes  through  them, 
enough   heat   is    generated   at  the   junction, 

where  the  resistance  is  greatest,  to  soften  and  weld  them  together. 
Fig.  353  shows  three  welded  joints  as  they  came  from  the  welder. 

1  If  H  is  the  heat  in  calories,  I  the  current  strength  in  amperes,  R  the  re- 
sistance in  ohms,  t  the  time  in  seconds,  and  0.24  the  number  of  calories  equiv- 
alent to  one  joule,  then  the  heat  equivalent  of  a  current  is 

H  =  0.24  X  I*Rt  calories. 


Fig.  353 


340 


ELECTRIC  CURRENTS 


V.  MAGNETIC  PROPERTIES   OF  A  CURRENT 

442.  Magnetic  Field  Around  a  Conductor.—  Dip  a  portion  of  a 
wire  carrying  a  heavy  current  into  fine  iron  filings.     A  thick  cluster 

of  them  will  adhere  to  the 

liiiiliiB^  wire  (Fig>  354) ;  they  wil1 

drop   off    as  soon   as  the 
Fig.  354  circuit  is  opened. 

The  experiment  shows  that  a  conductgrjh™"gh  whin.h 
an  electric  current  is  passing  lias  magnetic  properties. 
The  iron  filings  are  magnetized  by 
the  current  and  set  themselves  at 
right  angles  to  the  wire.  When  the 
circuit  is  broken,  they  lose  their  mag- 
netism and  drop  off. 

443.  Mapping   the   Magnetic    Field. 

—  Support  horizontally  a  sheet  of  cardboard 

or  of  glass  LB  with  a  hole  through  it.     Pass 

vertically  through  the  hole  a  wire,  W,  con- 
necting with  a  suitable  electric  generator, 

so  that  a  strong  current  can  be  sent  through 

the  circuit  (Fig.  355).     Close  the  circuit  and 

sift  iron  filings  on  the  paper  or  glass  about 

the  wire,  jarring  the  sheet  by  tapping  it.  The  filings  will  arrange  them- 
selves in  circular  lines  about  the 
wire.  Place  a  small  mounted  mag- 
netic needle  on  the  sheet  near  the 
wire;  it  will  set  itself  tangent  to 
the  circular  lines,  and  if  the  current 
is  flowing  downward,  the  north 
pole  will  point  in  the  direction  in 
which  the  hands  of  a  watch  move. 

The  lines  of  magnetic  force 
aboutTaT  wire  through  "which 
an"elecTric  current  is  flowing, 
Fig.  356  are   concentric  circles.     Fig. 


Fig.  355 


MAGNETIC  PROPERTIES   OF  A    CURRENT 


341 


356  was  made  from  a  photograph  of  these  circular  lines  of 
force  as  shown  by  iron  filings  on  a  plate  of  glass.  Their 
direction  relative  to  the  cur- 
rent is  given  by  the  following 
rule : 


G-rasp  the  wire  by  the  right 
hand  so  that  the  extended 
thumb  points  in  the  direction 
of  the  current;  then  the  fin- 
gers wrapped  around  the  wire 
indicate  the  direction  of  the  lines  of  force  (Fig.  357). 


Fig.  357 


Fig.  358  is  a  sketch  intended  to  show  the  direction  of 
these  circular  lines  of  magnetic  force  (^o^jnagnetic  whirl) 
which  everywhere  surround  a  wire  conveying  a  current. 


Fig.  358 


444.  Properties  of  a  Circular  Conductor.  —  Bend  a  copper 
wire  into  the  form  shown  in  Fig.  359,  the  diameter  of  the  circle  being 
about  20  cm.  Suspend  it  by  a  long  untwisted  thread,  so  that  the  ends 
dip  into  the  mercury  cups  shown  in  cross  section  in  the  lower  part  of 
the  figure.  Send  a  current  through  the  suspended  wire  by  connecting 
a  battery  to  the  binding  posts.  A  bar  magnet  brought  near  the  face 
of  the  circular  conductor  will  cause  the  latter  to  turn  about  a  vertical 
axis  and  take  up  a  position  with  its  plane  at  right  angles  to  the  axis 
of  the  magnet.  With  a  strong  current  the  circle  will  turn  under  the 
influence  of  the  earth's  magnetism. 

This  experjiiKmt_sJ:iow^  acts  like 

a   disk    magnet,  whose    poles   are    its   faces.     The  lines 


342 


ELECTRIC  CURRENTS 


of  force  surrounding  the  conductor  in  this  form  pass 
through  the  circle  and  around  from  one  face  to  the  other 
through  the  air  outside  the  loop. 
The  north-seeking  side  is  the  one 
from  which  the  lines  issue;  and  to 
an  observer  looking  toward  this  side, 
the  current  flows  around  the  loop 
counter-clockwise  (Fig.  360). 

If   instead   of   a   single  turn  we 
take  a  long  in- 
sulated wire  and 
coil    it    into     a 
number  of   par- 
allel circles  close 
together,  the 
magnetic    effect 
will  be  increased. 
Such  a  coil    is 
called  a  helix  or  solenoid ;  and  the  passage  of  an  electric 
current  through  it  gives  to  it  all  the  properties  of  a  cylin- 
drical bar  magnet. 

445.  Polarity  of  a  Helix. —  The  polarity  of  a  helix  may 
be  determined  by  the  following  rule: 

Crrasp  the  coil  with  the  right  hand  so  that  the  fingers  point 
in  the  direction  of  the  current ;  the  north  pole  will  then  be  in 
the  direction  of  the  extended  thumb. 

446.  Mutual  Action  of  Two  Currents.  —  Make  a  rectangular 
coil  of  insulated  copper  wire  by  winding  four  or  five  layers  around 
the   edge   of   a   board  about  25  cm.  square.     Slip  the  wire  off  the 
board  and  tie  the  parts  together  in  a  number  of  places  with  thread. 
Bend  the  ends  at  right  angles  to  the  frame,  remove  the  insulation, 


Fig.  359 


MAGNETIC  PROPERTIES   OF  A   CURRENT 


343 


and  give  them   the   shape  shown   in   Fig.   361.     Suspend  the  wire 
frame  by  a  long  thread  so  that  the  ends  dip  into  the  mercury  cups. 

Make  a  second  similar  but  smaller 
coil  and  connect  it  in  the  same  circuit 
with  the  rectangular  coil  and  a  battery. 

First.  Hold  the  coil  HK  with  its 
plane  perpendicular  to  the  plane  of  the 
coil  EF,  with  its  edge  H  parallel  to  F, 
and  with  the  currents  in  these  two 
adjacent  portions  flowing  in  the  same 
direction.  The  suspended  coil  will  turn 
upon  its  axis,  the  edge  F  approaching 
H,  as  if  it  were  attracted. 

Second.  Reverse  HK  so  that  the  cur- 
rents in  the  adjacent  portions  K  and  F 
flow  in  opposite  directions.  The  edge  F 
of  the  suspended  coil  will  be  repelled  by  K. 

Third.  Hold  the  coil  HK  within  EF,  so  that  their  lower  sides  form 
an  angle.  EF  will  turn  until  the  currents  in  its  lower  side  are  par- 
allel with  those  in  H,  and  flowing  in  the  same  direction. 

These  facts  may  be  summarized  in  the  following  laws 

of  action  between  cur- 


Fig.  361 


-  362 


I.  Parallel      cur- 
rents flowing  in  the 
satne    direction    at- 
tract. 

II.  Parallel     cur- 
rents flowing  in  oppo- 
site directions  repel. 

III.  Currents 
making   an    a  ng  I  e 
with  each  other  tend 


to  become  parallel  and  to  flow  in  the  same  direction. 

447.   Magnetic  Fields  about  Parallel  Currents.  —  Fig.  362 
was  made  from  a  photograph  of  the  magnetic  field  about 


344 


ELECTRIC  CURRENTS 


two  parallel  currents  in  the  same  direction  perpendicular 
to  the   figure.     Many  of   these  lines  of  'force  surround 

both  wires,  and  it 
is  the  tension  along 
them  that  draws 
the  wires  together. 
Fig.  363  was  made 
from  a  photograph 
of  the  field  when 
the  currents  were 
in  opposite  direc- 
tions. The  lines  of 
force  are  crowded 
together  between 

the  wires,  and   their  reaction  in  their  effort  to  recover 
their  normal  position  forces  the  wires  apart. 


VI.    ELECTROMAGNETS 

448.  Effect  of  Introducing  Iron  into  a  Solenoid.  —  Wind 
evenly  on  a  paper  tube,  about  2  cm.  in  diameter  and  15  cm.  long, 
three  layers  of  No.  18  insulated  copper  wire.  Support  the  tube  and 
wire  in  a  slot  cut  in  a  sheet  of  cardboard.  Pass  an  electric  current 
through  the  solenoid  and  note  the  magnetic  field  as  mapped  out  by 
iron  filings  (Fig.  364).  Repeat 
after  filling  the  tube  with  straight 
soft  iron  wires.  The  magnetic 
field  will  be  greatly  strengthened 
by  the  iron. 

A  helix  of  wire  about  an 
iron  core  is  an  electromagnet. 
It  was  first  made  by  Sturgeon  Fig.  364 

in  1825.     The  presence  of 

the  iron  core  greatly  increases  the  number  of  lines  of  force 
threading  through  the  helix  from  end  to  end,  by  reason  of 


ELECTRON  A  GNET8 


345 


the  greater  permeability  of  iron  as  compared  with  air 
(Fig.  365).  If  the  iron  is  omitted,  there  are  not  only 
fewer  lines  of  force,  but  because  of  their  leakage  at  the 
sides  of  the  helix,  fewer  traverse  the  entire  length  of  the  coil. 


Fig.  365 

The  soft  iron  core  of  an  electromagnet  does  not  show 
much  magnetism  except  while  the  current  is  flowing 
through  the  magnetizing  coil.  The  loss  of  magnetism  is 
not  quite  complete  when 
the  current  is  interrupted; 
the  small  amount  remain- 
ing is  called  residual  mag- 
netism. 


449.  Relation  between  a 
Magnet  and  a  Flexible  Con- 
ductor. —  Iron  filings  arranged 
in  circles  about  a  conductor 
may  be  regarded  as  flexible  mag- 
netized iron  winding  itself  into 
a  helix  around  the  current ;  con- 
versely, a  flexible  conductor, 
carrying  a  current,  winds  itself 
around  a  straight  bar  magnet. 
The  flexible  conductor  of  Fig. 
366  may  be  made  of  tinsel  cord 
or  braid.  Directly  the  circuit 


Fig.  366 


346 


ELECTRIC  CURRENTS 


is  closed,  the  conductor  winds  slowly  around  the  vertical  magnet; 
if  the  current  is  then  reversed,  the  conductor  unwinds  and  winds 
up  again  in  the  reverse  direction. 

450.  The  Horseshoe  Magnet.  —  The  form  given  to  an 
electromagnet  depends  on  the  use  to  which  it  is  to  be  put. 
The  horseshoe  or  U-shape  (Fig.  367)  is  the  most  common. 

The  advantage  of  this  form  lies 
in  the  fact  that  all  lines  of  mag- 
netic force  are  closed  curves,  pass- 
ing through  the  core  from  the 
south  to  the  north  pole,  and  com- 
pleting the  circuit  through  the  air 
from  the  north  pole  back  to  the 
.south  pole.  The  U-shape  lessens 


Fig.  367 


the  distance  through  the  air  and  thus  increases  the  number 
of  lines.  Moreover,  when  an  iron  bar,  called  the  arma- 
ture, is  placed  across  the  poles,  the  air  gap  is  reduced 
to  a  thin  film,  the  number  of  lines  is  increased  to  a 
maximum  with  a  given  current  through  the  helix,  and 
the  magnet  exercises  the  greatest  pull  on  the  armature. 

When  the  armature  is  in  contact  with  the  poles,  the 
magnetic  circuit  is  %all  iron,  and  is  said  to  be  a  closed 
magnetic  circuit.  The  residual  magnetism  is  then  much 
greater  than  in  the  case  of  an  open  magnetic  circuit  with 
an  air  gap. 

Bring  the  armature  in  contact  with  the  iron  poles  of  the  core,  and 
close  the  electric  circuit ;  after  the  circuit  is  opened,  the  armature  will 
still  cling  to  the  poles  and  can  be  removed  only  with  some  effort. 
Then  place  a  piece  of  thin  paper  between  the  poles  and  the  armature. 
After  the  magnet  has  again  been  excited  and  the  circuit  opened,  the 
armature  will  not  now  "stick."  The  paper  makes  a  thin  air  gap 
between  the  poles  of  the  magnet  and  the  armature,  and  thus  reduces 
the  residual  magnetism. 


MEASURING  INSTRUMENTS 


347 


451.  Applications  of  Electromagnets. 
—  The  uses  to  which  electromagnets  are  put 
in  the  applications  of  electricity  are  so  nu- 
merous that  a  mere  reference  to  them  must 
suffice.  The  electromagnet  enters  into  the 
construction  of  electric  bells,  telegraph  and 
telephone  instruments,  dynamos,  motors, 
signaling  devices,  etc.  It  is  also  extensively 
used  in  lifting  large  masses  of  iron,  such 
as  castings,  rolled  plates,  pig  iron,  and  steel 
girders  (Fig.  368).  The  lifting  power  de- 
pends chiefly  on  the  cross  section  of  the  iron 
core  and  on  the  ampere  turns;  that  is,  on  the 
product  of  the  number  of  amperes  of  cur- 
rent and  the  number  of  turns  of  wire  wound 
on  the  magnet. 


Fig.  368 


VII.    MEASURING  INSTRUMENTS 

452.  The  Galvanometer.  — The  instrument  for  the  com- 
parison of  currents  by  means  of  their  magnetic  effects  is 
called  a  galvanometer.     A  galvanoscope  (§  413)  becomes 
a  galvanometer  by  providing  it  with  a  scale  so  that  the 
deflections   may   be   measured.     If   the   galvanometer  is 
calibrated,  so  as  to  read  directly  in  amperes,  it  is  called  an 
ammeter.     In  very  sensitive  instruments  a  small  mirror  is 
attached  to  the  movable  part  of  the  instrument ;  it  is  then 
called  a  mirror  galvanometer.     Sometimes  a  beam  of  light 
from  a  lamp  is  reflected  from  this  small  mirror  back  to  a 
scale,  and  sometimes  the  light  from   a  scale  is  reflected 
back  to  a  small  telescope,  by  means  of  which  the  deflections 
are  read.     In  either  case  the  beam  of  light  then  becomes  a 
long  pointer  without  weight. 

453.  The  d'Arsonval  Galvanometer.  —  One  of  the  most 
useful  forms  of   galvanometer   is   the   d'Arsonval.     The 


348 


ELECTRIC  CURRENTS 


Fig.  369 


Fig.  370 


plan  of  it  is  shown  in  Fig.  369  and  a  complete  working 
instrument  in  Fig.  370.  Between  the  poles  of  a  strong 
permanent  magnet  of  the  horseshoe  form  swings  a  rec- 
tangular coil  of  fine  wire  in  such  a  way  that 

the  current  is  led 

into    the    coil    by 

the  fine  suspending 

wire,  and   out   by 

the  wire  spiral  run- 
ning   to  the  base. 

A  small  mirror  is 

attached  to  the 

coil  to  reflect  light 

from  a  lamp  or  an 

illuminated    scale. 

Sometimes  the  coil 

carries  a  light  alum- 
inum pointer,  which  traverses  a  scale.  Inside  the 
coil  is  a  soft,  iron  tube  supported  from  the  back  of  the 
case.  It  is  designed  to  concentrate  the  lines  of  force  in 
the  narrow  openings  between  it  and  the  poles  of  the 
magnet. 

In  the  d'Arsonval  galvanometer  the  coil  is  movable  and 
the  magnet  is  fixed.  Its  chief  advantages  are  simplicity 
of  construction,  comparative  independence  of  the  earth's 
magnetic  field,  and  the  quickness  with  which  the  coil 
comes  to  rest  after  deflection  by  a  current  through  it. 

454.  The  Voltmeter.  —  The  voltmeter  is  an  instrument 
designed  to  measure  the  difference  of  potential  in  volts. 
For  direct  currents  the  most  convenient  portable  volt- 
meter is  made  on  the  principle  of  the  d'Arsonval  galva- 
nometer. The  appearance  of  one  of  the  best-known 


MEASURING  INSTRUMENTS 


349 


instruments  of  this  class  is  shown  in  Fig.  371.  The 
interior  is  represented  by  Fig.  372,  where  a  portion  of  the 
instrument  is  cut  away 
to  show  the  coil  and 
the  springs.  The  cur- 
rent is  led  in  by  one 
spiral  spring  and  out 
by  the  other.  At- 
tached to  the  coil  is 
a  very  light  aluminum 
pointer,  which  moves 
over  the  scale  seen  in 

Fig.     371,    where     it  ^  3j 

stands  at  zero.     Soft 

iron  polepieces  are  screwed  fast  to  the  poles  of  the 
permanent  magnet,  and  they  are  so  shaped  that  the 
divisions  of  the  scale  in  volts  are  equal. 

In  circuit  with  the  coil  of  the  instrument  is  a  coil  of  wire 

of  high  resistance,  so  that 
when  the  voltmeter  is 
placed  in  circuit,  only  a 
small  current  will  flow 
through  it. 

455.  The  Ammeter,  de- 
signed to  measure  electric 
currents  in  amperes,  is  very 
similar  in  construction  to 
the  voltmeter.  Its  coil  has 
only  a  few  turns  of  wire 

and  its  resistance  is  low,  so  that  when  the  ammeter  is 
placed  in  circuit,  it  will  not  change  the  value  of  the  cur- 
rent to  be  measured. 


350  ELECTRIC  CURRENTS 

456.  Divided  Circuits  —  Shunts.  —  When  the  wire  leading 
from  any  electric  generator  is  divided  into  two  branches, 
as  at  B  (Fig.  373),  the  current  also  divides,  part  flowing 

by  one   path  and   part   by 
the  other.    The  sum  of  these 
two  currents  is  always  equal 
to  the  current  in  the  undi- 
vided  part   of  the    circuit, 
since  there  is  no  accumula- 
tion of  electricity  at  any  point.     Either  of  the  branches  be- 
tween B  and  A  is  called  a  shunt  to  the  other,  and  the  currents 
through  them  are  inversely  proportional  to  their  resistances. 

457.  Resistance  of  a  Divided  Circuit.  —  Let  the  total  resist- 
ance between  the  points  A  and  B  (Fig.  373)  be  represented  by  R, 
that  of  the  branch  BmA  by  r,  a-nd  of  En  A  by  r.r     The  conductance  of 
BA  equals  the  sum  of  the  conductances  of  the  two  branches ;  and,  as 
conductance  is  the  reciprocal  of  resistance,  the  conductances  of  BA, 

BmA,  and  J5n^4  are  — ,    -,  and  —   respectively;  then— =  — +  — . 
R      r  r  K,      r      r 

From  this  we  derive  R  =    rr     .    To  illustrate,   let   a  galvanometer 

r  +  r' 

whose  resistance  is  100  ohms  have  its  binding  posts  connected  by  a 
shunt  of  50  ohms  resistance;  then  the  total  resistance  of  this  divided 

I  r\r\  y   Kf\ 

circuit  is  ±^2L^  -  33 1  ohms.     The  introduction  of  a  shunt  always 
100  +  50 

lessens  the  resistance  between  the  points  connected. 

458.  Loss  of  Potential  along  a  Conductor.  —  When  a  cur- 
rent flows   through   a  conductor   a   difference  of  poten- 
tial exists,  in  general,    between    different    points  on  it. 
Let  A,  B,   Q  be  three  points  on  a  conductor  conveying 
a  current,   and   let  there    be   no   source   of  E.  M.  F.    be- 
tween these  points.     Then  if  the  current  flows  from  A  to 
B,  the  potential   at  A  is  higher  than  at  B,  and  the  poten- 
tial at  B  is  higher  than  at  O.     If  the  potential  difference 


MEASURING  INSTRUMENTS 


351 


between  A  and  B  and  that  between  B  and  0  be  measured, 
the  ratio  of  the  two  will  be  the  same  as  the  ratio  of 
the  resistances  between  the  same  points.  This  is  only 

TjJ 

another  statement  of  Ohm's  law.     For  since  /=  — ,  and  the 

H 

current  is  the  same  through  the  two  adjacent  sections  of 
the  conductor,  the  ratio  of  the  potential  differences  to  the 
resistances  of  the  two  sections  is  the  same.  This  impor- 
tant principle,  of  which  great  use  is  made  in  electrical 
measurements,  may  be  expressed  by  saying  that,  when  the 
current  is  constant,  the  loss  of  potential  along  a  conductor 
is  proportional  to  the  resistance  passed  over. 

459.  Wheatstone's  Bridge.  —  The  Wheatstone's  Bridge  is  a  de- 
vice for  measuring  resistances.  The  four  conductors  /2i,  R2,  R3,  R4 
are  the  arms  and  BD  the  bridge  (Fig. 
374).  When  the  circuit  is  closed  by 
closing  the  key  K^  the  current  divides 
at  A,  the  two  parts  reuniting  at  C. 
The  loss  of  potential  along  ABC  is  the 
same  as  along  ADC.  If  no  current 
flows  through  the  galvanometer  G 
when  the  key  KI  is  also  closed,  then 
there  is  no  potential  difference  be- 
tween B  and  D  to  produce  a  current. 
Under  these  conditions  the  loss  of  po- 
tential from  A  to  B  is  the  same  as 
from  A  to  D.  We  may  then  get  an 


Fig.  374 


expression  for  these  potential  differences  and  place  them  equal  to 
each  other. 

Let  /i  be  the  current  through  7?i ;  it  will  also  be  the  current 
through  R±,  because  none  flows  across  through  the  galvanometer. 
Also  let  72  be  the  current  through  the  branch  ADC.  Then  the  poten- 
tial difference  between  A  and  B  by  Ohm's  law  (§  435)  is  equal  to  R±I\ ; 
and  the  equal  potential  difference  between  A  and  D  is  R^I2.  Equating 
these  expressions,  jRi/i  =  R2fz («) 

In  the  same  way  the  equal  potential  differences  between  B  and  C 


852  ELECTRIC  CURRENTS 

and  D  and  C  give  RJi  =  RsI2    ...........      (b) 

Dividing  (a)  by  (6)  gives 


(Equation    37) 


In  practice  three  of  the  four  resistances  are  adjustable  and  of 
known  value.  They  are  adjusted  until  the  galvanometer  shows  no 
deflection  when  the  key  KI  is  closed  after  key  K*.  The  value  of  the 
fourth  resistance  is  then  derived  from  the  relation  in  equation  (37). 


Questions  and  Problems 

1.  An  electric  bell  wire  passes  through  a  room  and  the  battery 
is  inaccessible.     How  may  one  determine  the  direction  of  the  current 
through  the  wire? 

2.  How  can  it  be  proved  that  the  strength  of  current  is  the  same 
in  all  portions  of  an  undivided  circuit? 

3.  The  poles  of  a  battery  are  joined  by  a  thin  platinum  wire, 
which  is  heated  to  a  dull  red.     If  a  piece  of  ice  is  applied  to  the 
wire  at  one  end,  the  remainder  of  the  wire  will  glow  more  brightly. 
Explain. 

4.  Given  a  charged  storage  battery.      Determine  which  is  the 
positive  pole. 

5.  While  a  current  is  passing  through  a  helix,  a  small  iron  rod 
is  brought  near  one  end  of  the  helix  and  in  line  with  its  axis.     The 
iron  rod  will  be  drawn  into  the  helix.     Explain. 

6.  A  current  passing  through  a  long  elastic  spiral  of  wire  causes 
it  to  shorten.     Explain. 

7.  Calculate  the  resistance  of  100  ft.  of  copper  wire  (k  =  10.19) 
No.  24  (diam.  =  0.0201  in.). 

8.  No.  20  wire  has  a  diameter  of  0.032  in.     How  many  feet  of 
German  silver  wire  (k  =  181.3)  will  it  take  to  make  a  20-ohm  coil? 

9.  How  many  feet  of  iron  wire  (k  =  61.3),  No.  10  (diam.  =  0.1014 
in),  will  it  take  to  make  a  coil  of  50  ohms  resistance? 

10.  A  current  of  one  ampere  deposits  by  electrolysis  1.1833  gm. 
of  copper  in  an  hour.  How  many  amperes  in  10  hours  will  deposit 
l.kgm.  of  copper? 


QUESTIONS  AND  PROBLEMS  353 

11.  A  current  of  0.5  ampere  is  passed  through  a  solution  of  silver 
nitrate  for  30  min.     How  much  silver  is  deposited? 

12.  What  strength  of  current  in  amperes  will  deposit  10  gm.  of 
silver  by  electrolysis  in  an  hour? 

13.  What  current  will  a  battery  having  an  E.  M.  F.  of  2.2  volts  and 
an  internal  resistance  of  0.2  ohm  supply  through  an  external  resist- 
ance of  5  ohms? 

14.  How  large  a  current  will  a  battery  of  6  cells  (E.  M.  F.,  1.5  volts 
each)  connected  in  series  send  through  an  external  resistance  of  6 
ohms,  the  internal  resistance  of  each  cell  being  0.5? 

15.  If  a  dry  cell  has  an  E.  M.  F.  of  1.5  volts  and  sends  a  current  of 
20  amperes  through  an  ammeter  (resistance  negligible),  what  is  the 
internal  resistance  of  the  cell? 

16.  What  current  will  be  derived  from  a  Daniell  cell,  E.  M.  F.  1.1 
volts,  internal  resistance  1  ohm,  in  series  with  a  dry  cell,  E.  M.  F.  1.5 
volts,  internal  resistance  0.2  ohm,  when  the  external  resistance  is  4 
ohms  ? 

17.  A  current  of  10  amperes  passes  through  a  resistance  of  1  ohm 
for  half  an  hour.     How  many  calories  of  heat  are  generated? 

18.  A  current  of  2.1  amperes  is  sent  through  a  divided  circuit  of 
two  branches,  with  resistances  of  5  and  10  ohms  respectively.     Cal- 
culate the  current  in  each  branch. 

19.  If  the  current  through  an  incandescent  lamp  is  0.55  ampere 
and  the  potential  difference  between  its  terminals  110  volts,  what  is 
the  resistance  of  the  lamp? 

20.  What  resistance  would  be  necessary  in  circuit  with  an  electric 
lamp  when  the  potential  difference  between  its  terminals  is  50  volts, 
the  pressure  in  the  main  line  200  volts,  and  the  current  through  the 
lamp  12  amperes? 


CHAPTER  XIII 

ELECTROMAGNETIC  INDUCTION 

I.  FARADAY'S  DISCOVERIES 

460.   Electromotive  Force  Induced  by  a  Magnet.  —  Wind  a 

number  of  turns  of  fine  insulated  wire  around  the  armature  of  a  horse- 
shoe magnet,  leaving  the  ends  of  the  iron  free  to  come  in  contact  with 
the  poles  of  the  permanent  mag- 
net. Connect  the  ends  of  the 
coil  to  a  sensitive  galvanometer, 
the  armature  being  in  contact 


with  the  magnetic  poles,  as  shown 
in  Fig.  375.     Keeping  the  mag- 
net fixed,  suddenly  pull  off  the  armature, 
show  a  momentarv  current. 


Fig.  375 

The  galvanometer  will 
Suddenly  bring  the  armature  up  to  the 
poles  of  the  magnet;  an- 
other momentary  current 
in  the  reverse  direction 
will  flow  through  the  cir- 
cuit. 

Connect  a  coil  of  insu- 
lated wire,  consisting  of  a 
large  number  of  turns,  in 
circuit  with  a  d'Arsonval 
galvanometer  (Fig.  376). 
Thrust  quickly  into  the 
coil  the  north-  pole  of  a  bar 
magnet.  The  galvanome- 
ter will  show  a  transient 

current,    which    will    flow 

only  during  the  motion  of 
the    magnet.     When    the 

magnet  is  suddenly  withdrawn  a  transient  current  is  pro'duced  in  the 
opposite  direction  to  the  first  one.     If  the  south  pole  be  thrust  into 

354 


FARADAY'S  DISCOVERIES  355 

the  coil,  and  then  withdrawn,  the  currents  in  both  cases  are  the  reverse 
of  those  with  the  north  pole.  If  we  substitute  a  helix  of  a  smaller 
number  of  turns,  or  a  weaker  bar  magnet,  the  deflection  will  be  less. 

The  momentary  electromotive  forces  generated  in  the 
coil  are  known  as  induced  electromotive  forces,  and  the  cur- 
rents as  induced  currents.  They  were  discovered  by 
Faraday  in  1831. 

461.  Laws  of  Electromagnetic  Induction. —  When  the  arma- 
ture in  the  first  experiment  of  the  last  article  is  in  contact 
with  the  poles  of  the  magnet,  the  number  of  lines  of  force 
passing  through  the  coil,  or  linked  with  it,  is  a  maximum. 
When  the  armature  is  pulled  away,  the  number  of  mag- 
netic lines  threading  through  the  coil  rapidly  diminishes. 

When  the  magnet  in  the  second  experiment  is  thrust 
into  the  coil,  it  carries  its  lines  of  force  with  it,  so  that 
some  of  them  at  least  encircle,  or  are  linked  with,  the  wires 
of  the  coil.  In  both  experiments  an  electromotive  force 
is  generated  only  while  the  number  of  lines  so  linked  with 
the  coil  is  changing.  The  E.  M.  F.  is  generated  in  the 
coil  in  accordance  with  the  following  laws: 

I.  An  increase  in  the  number  of  lines  of  force  threading 
through  a  coil  produces  an  indirect  electromotive  force; 
a  decrease  in  the  number  of  lines  produces  a  direct  electro- 
motive force. 

II.  The  electromotive  force  induced  is  proportional  to  the 
rate  of  change  in  the  number  of  lines  of  force  threading 
through  the  coil. 

These  two  laws  give  the  direction  and  value  of  induced 
electromotive  forces.  A  direct  E.  M.  F.  has  a  clockwise 
direction  to  an  observer  looking  along  the  lines  of  force  of 
the  magnet;  an  indirect  E.  M.  F.  is  one  in  the  opposite 


356 


ELECTR  OMA  GNETIC  IND  UCTION 


\ 


Fig.  377 


direction.  Thus,  in  Fig.  377  the  north  pole  of  the  mag- 
net is  moving  into  the  corl~fn 
the  direction  of  the  arrow; 
there  is  an  increase  in  the 
number  of  lines  passing 
through  the  coil,  and  the 
E.  M.  F.  and  current  are  in- 
direct or  opposite  watch 
hands,  as  shown  by  the 
arrows  on  the  coil,  to  an  ob- 
server looking  at  the  coil  in  the  direction  of  the  arrow  on 
the  magnet. 

462.   Induction  by  Currents.  —  Connect  the  coil  of  Fig.  376  to  a 
d'Arsonval  galvanometer,  and  a  second  smaller  coil  to  the  terminals 

of  a  battery  (Fig.  378).  If  the 
current  through  P  is  kept  con- 
stant, when  P  is  made  to  ap- 
proach S  an  E.  M.  F.  is  gener- 
ated in  S  tending  to  sejid  a 
current  in  a  direction  opposite 
to  the  current  around  P;  re- 
moving the  coil  P  generates  an 
opposite  E.  M.  F.  These  E.  M. 
F.'s  act  in  S  only  so  long  as  P 
is  moving. 

Next  insert  the  coil  P  in  S 
with  the  battery  circuit  open. 
If  then  the  battery  circuit  is 
closed,  the  needle  of  the  galva- 
nometer will  be  deflected,  but 
will  shortly  come  again  to  rest 
at  zero.  The  direction  of  this 
momentary  current  is  opposite 
to  that  in  P.  Opening  the  bat- 
tery circuit  produces  another  momentary  current  through  S  but  in  the 
opposite  direction.  Increasing  and  decreasing  the  current  through  P 
has  the  same  effect  as  closing  and  opening  the  circuit. 


Fig.  378 


FAEADAY'S  DISCOVERIES  357 

If  while  P  is  inside  S  with  the  battery  circuit  closed,  a  bar  of  soft 
iron  is  placed  within  P,  there  is  an  increase  of  magnetic  lines  through 
both  coils  and  the  inductive  effect  in  <S  is  the  same  as  that  produced 
by  closing  the  circuit  through  P. 

The  coil  P  is  called  the  primary  and  iS  the  secondary 
coil.  The  results  may  be  summarized  as  follows: 

I.  Momentary  indirect  electromotive  forces   are  in- 
duced in  the  secondary  by  the  approach,  the  starting,  or 
the  strengthening  of  a  current  in  the  primary  coil. 

II.  Momentary  direct  electromotive  forces  are  induced 
in  the  secondary  by  the  receding,  the  stopping,  or  the 
weakening  of  a  current  in  the  primary  coil. 

The  primary  coil  becomes  a  magnet  when  carrying  an 
electric  current  (§  444)  and  acts  toward  the  secondary  coil 
as  if  it  were  a  magnet.  The  soft  iron  increases  the  mag- 
netic flux  through  the  coil  and  so  increases  the  induction. 

463.  Lenz's  Law.  —  When  the  north  pole  of  the  magnet  is 
thrust  into  the  coil  of  Fig.  3? 7,  the  induced  current  flowing 
in  the  direction  of  the  arrows  produces  lines  of  force  run- 
ning in  the  opposite  direction  to  those  from  the  magnet 
(§  443).  These  lines  of  force  tend  to  oppose  the  change 
in  the  magnetic  field  within  the  coil,  or  the  magnetic  field 
set  up  by  the  coil  opposes  the  motion  of  the  magnet. 

Again,  when  the  primary  coil  of  Fig.  378  is  inserted 
into  the  secondary,  the  induced  current  in  the  latter  is 
opposite  in  direction  to  the  primary  current,  and  parallel 
currents  in  opposite  directions  repel  each  other.  In  every 
case  of  electromagnetic  induction  the  change  in  the  mag- 
netic field  which  produces  the  induced  current  is  always 
opposed  by  the  magnetic  field  due  to  the  induced  current 
itself. 

The  law  of  Lenz  respecting  the  direction  of  the  induced 
current  is  broadly  as  follows : 


358 


EL  ECTR  ON  A  GNETIC  IND  UCTION 


The  direction  of  an  induced  current  is  always  such  that 
it  produces  a  magnetic  field  opposing  the  motion  or  change 
which  induces  the  current. 

II.    SELF-INDUCTION 

464.  Joseph  Henry's  Discovery.  —  Joseph  Henry  discov- 
ered that  a  current  through  a  helix  with  parallel  turns 
acts  inductively  on  its  own  circuit,  producing  what  is 
often  called  the  extra  current,  and  a  bright  spark  across 
the  gap  when  the  circuit  is  opened.     The  effects  are  not 
very  marked  unless  the  helix  contains  a  soft  iron  core. 

Let  a  coil  of  wire  be  wound 
around  a  wooden  cylinder  (Fig. 
379).  When  a  current  is  flowing 
through  this  coil,  some  of  the  lines 
of  force  around  one  turn,  as  A, 
thread  through  adjacent  turns;  if 
the  cylinder  is  iron,  the  number  of 
lines  threading  through  adjacent 
turns  will  be  largely  increased  on 
account  of  the  superior  permeability  of  the  iron  (§  365). 
Hence,  at  the  make  of  the  circuit,  the  production  of  mag- 
netic lines  threading  through  the  parallel  turns  of  wire 
induces  a  counter-E.  M.  F.  opposing  the  current.  The 
result  is  that  the  current  does  not  reach  at  once  the  value 
given  by  Ohm's  law.  At  the  break  of  the  circuit,  the  in- 
duction on  the  other  hand  produces  a  direct  E.  M.  F.  tend- 
ing to  prolong  the  current.  With  many  turns  of  wire,  this 
direct  E.  M.  F.  is  high  enough  to  break  over  a  short  gap 
and  produce  a  spark. 

465.  Illustrations  of  Self -Induction.  —  Connect  two  or  three  cells 
in  series.     Join  electrically  a  flat  file  to  one  pole  and  a  piece  of  iron 
wire  to  the  other.     Draw  the  end  of  the  wire  lengthwise  along  the 


Fig.  379 


Joseph  Henry  (1797-1878)  was  born  at  Albany,  New  York. 
The  reading  of  Gregory's  Lectures  on  Experimental  Philosophy 
interested  him  so  greatly  in  science  that  he  began  experimenting. 
In  1829  he  constructed  his  first  electromagnet.  In  1832  he  was 
appointed  professor  of  natural  philosophy  at  Princeton  College. 
In  1846  he  became  secretary  of  the  Smithsonian  Institution  in 
Washington.  It  is  almost  certain  that  he  anticipated  Faraday's 
great  discovery  of  magneto-electric  induction  by  a  whole  year 
but  failed  to  announce  it.  His  principal  investigations  were  in 
electricity  and  magnetism,  and  chiefly  in  the  realm  of  induced 
currents. 


THE  INDUCTION  COIL 


359 


|  ,     M     r- 

B 

4 

i 

i  i 

i 

i  i  —  ' 

file ;  some  sparks  will  be  visible,  but  they  emit  little  light.  Now  put 
an  electromagnet  in  the  circuit  to  increase  the  self-induction;  the 
sparks  are  now  much  longer  and  brighter. 

Connect  as  shown  in  Fig.  380  a  large  electromagnet  M,  a  storage 
battery  B,  a  circuit  breaker  K,  and  an  incandescent  lamp  L  of  such  a 
size  that  the  battery  alone  will  light  it  to 
nearly  its  full  candle  power.  The  circuit 
divides  between  the  lamp  and  the  electro- 
magnet, and  since  the  latter  is  of  low  resist- 
ance, when  the  current  reaches  its  steady 
state  most  of  it  will  go  through  the  coils  of 
the  magnet,  leaving  the  lamp  at  only  a  dull 
red.  At  the  instant  when  the  circuit  is 
closed,  the  self-induction  of  the  magnet  acts 
against  the  current  and  sends  most  of  it 
around  through  the  lamp.  It  accordingly 
lights  up  at  first,  but  quickly  grows  dim  as 
the  current  rises  to  its  steady  value  in  M. 

Now  open  the  circuit  breaker  Kt  cutting 
off  the  battery.     The  only  closed  circuit  is  Fi£-  38° 

now  the  one  through  the  magnet  and  the  lamp;  but  the  energy 
stored  in  the  magnetic  field  of  the  electromagnet  is  then  converted 
into  electric  energy  by  means  of  self-induction,  and  the  lamp  again 
lights  up  brightly  for  a  moment. 

III.    THE  INDUCTION  COIL 

466.  Structure  of  an  Induction  Coil.  —  The  induction  coil 
is  commonly  used  to  give  transient  flashes  of  high  electro- 
motive force  in  rapid  suc- 
cession. A  primary  coil  of 
comparatively  few  turns  of 
stout  wire  is  wound  around 
an  iron  core,  consisting  of 
a  bundle  of  iron  wires  to 
avoid  induced  or  eddy  cur- 
rents in  the  metal  of  the 
Fig.  381  core;  outside  of  this,  and 


360 


ELECTROMAGNETIC  INDUCTION 


carefully  insulated  from  it,  is  the  secondary  of  a  very 
large  number  of  turns  of  fine  wire.  The  inner  or  primary 
coil  is  connected  to  a  battery  through  a  circuit  breaker 
(Fig.  381).  This  is  an  automatic  device  for  opening  and 
closing  the  primary  circuit  and  is  actuated  by  the  magnetism 
of  the  iron  core.  At  the  "  make  "  and  "  break  "  of  the  pri- 
mary circuit  electromotive  forces  are  induced  in  the  sec- 
ondary in  accordance  with  the  laws  of  electromagnetic 
induction  (§  461).  Large  induction  coils  include  also  a 
condenser.  It  is  placed  in  the  base  and  consists  of  two 
sets  of  interlaid  layers  of  tin  foil,  separated  b}^  sheets  of 
paper  saturated  with  paraffin.  The  two  sets  are  connected 
to  two  points  of  the  primary  circuit  on  opposite  sides  of 
the  circuit  breaker  (Fig.  382). 


J\  A  J\  A  J\  A  J\  A  J\  A  J\  A  A  A  J\ 

P 

f^ 

L 

i  &  n  n 

u  u  VTA)  vru  u  v)  O  vru  vrvl  \r\3 

P 

=p  - 

- 

T 

h     \m 


Fig.  382 

467.  Action  of  the  Coil.  —  Figure  382  shows  the  arrange- 
ment of  the  various  parts  of  an  induction  coil.  The  cur- 
rent first  passes  through  the  heavy  primary  wire  PP, 
thence  through  the  spring  A,  which  carries  the  soft  iron 
block  F,  then  across  to  the  screw  &,  and  so  back  to  the 
negative  pole  of  the  battery.  This  current  magnetizes 
the  iron  core  of  the  coil,  and  the  core  attracts  the  soft  iron 


THE  INDUCTION  COIL  361 

block  F,  thus  breaking  the  circuit  at  the  point  of  the 
screw  b.  The  core  is  then  demagnetized,  and  the  release 
of  F  again  closes  the  circuit.  Electromotive  forces  are 
thus  induced  in  the  secondary  coil  SS,  both  at  the  make 
and  the  break  of  the  primary.  The  high  E.M.F.  of  the 
secondary  is  due  to  the  large  number  of  turns  of  wire  in 
it  and  to  the  influence  of  the  iron  core  in  increasing  the 
number  of  lines  of  force  which  pass  through  the  entire 
coil. 

The  self-induction  of  the  primary  has  a  very  important 
bearing  on  the  action  of  the  coil.  At  the  instant  the  cir- 
cuit is  closed,  the  counter  E.  M.  F.  opposes  the  battery 
current,  and  prolongs  the  time  of  reaching  its  greatest 
strength.  Consequently  the  E.  M.  F.  of  the  secondary 
coil  will  be  diminished  by  self-induction  in  the  primary. 
The  E.  M.  F.  of  self-induction  at  the  "break"  of  the  pri- 
mary is  direct,  and  this  added  to  the  E.  M.  F.  of  the  battery 
produces  a  spark  at  the  break  points  of  the  circuit  breaker. 

468.  Action  of  the  Condenser.  — The  addition  of  a  con- 
denser increases  the  E.  M.  F.  of  the  secondary  coil  in  two 
ways  :  1.  It  gives  such  an  increase  of  capacity  to  the 
primary  coil  that  at  the  moment  of  breaking  the  circuit 
the  potential  difference  between  the  contact  points  does 
not  rise  high  enough  to  cause  a  spark  discharge  across  the 
air  gap.  The  interruption  of  the  primary  is  therefore 
more  abrupt,  and  the  E.  M.  F.  of  the  secondary  is  in- 
creased. 2.  After  the  break,  the  condenser  (7,  which  has 
been  charged  by  the  E.  M.  F.  of  self-induction,  discharges 
back  through  the  primary  coil  and  the  battery.  The  con- 
denser causes  an  electric  recoil  in  the  current,  and  returns 
the  stored  charge  as  a  current  in  the  reverse  direction 
through  the  primary,  thus  demagnetizing  the  core,  in- 


362  ELECTROMAGNETIC  INDUCTION 

creasing  the  rate  of  change  of  magnetic  flux,  and  increas- 
ing the  induced  E.  M.  F.  in  the  secondary.  The  condenser 
momentarily  stores  the  energy  represented  by  the  spark 
when  no  condenser  is  used,  and  then  returns  it  to  the 
primary  and  by  mutual  induction  to  the  secondary,  as  in- 
dicated by  the  longer  spark  or  the  greater  current.  When 
the  secondary  terminals  are  separated,  the  discharge  is  all 
in  one  direction  and  occurs  when  the  primary  current  is 
broken. 

469.  Experiments  with  the  Induction  Coil.  —  1.  Physiological 
Effects.  —  Hold  in  the  hands  the  electrodes  of  a  very  small  induction 
coil,  of  the  style  used  by  physicians.  When  the  coil  is  working,  a 
peculiar  muscular  contraction  is  produced. 

The  "shock"  from  large  coils  is  dangerous  on  account 
of  tbe  high  E.  M.  F.  The  danger  decreases  with  the  in- 
crease in  the  rapidity  of  the  impulses  or  alternations. 
Experiments  with  induction  coils,  worked  by  alternating 
currents  of  very  high  frequency,  have  demonstrated  that 
the  discharge  of  the  secondary  may  be  taken  through  the 
body  without  injury. 

2.  Mechanical  Effects.  —  Hold  a  piece  of  cardboard  between  the 
electrodes  of  an  induction  coil  giving  a  spark  3  cm.  long.     The  card 
will  be  perforated,  leaving  a  burr  on  each  side.     Thin  plates  of  any 
nonconductor  can  be  perforated  in  the  same  manner. 

3.  Chemical  Effects.  —  Place  on  a  plate  of  glass  a  strip  of  white 
blotting-paper  moistened  with  a  solution  of  potassium  iodide  (a  com- 
pound of  potassium  and  iodine)  and  starch  paste.     Attach  one  of  the 
electrodes   of   a   small    induction    coil  to   the  margin  of   the  paper. 
Handle  a  wire  leading  to  the  other  electrode  with  an  insulator,  and 
trace  characters  with  the  wire  on  the  paper  when  the  coil  is  in  action. 
The  discharge  decomposes  the  potassium  iodide,  as  shown  by  the  blue 
mark.    This  blue  mark  is  due  to  the  action  of  the  iodine  on  the  starch. 

If  the  current  from  the  secondary  of  an  induction  coil  be  passed 
through  air  in  a  sealed  tube,  the  nitrogen  and  oxygen  will  combine  to 


THE  INDUCTION  COIL 


363 


Fig.  383 


form  nitrous  acid.    This  is  the  basis  of  some  of  the  commercial  methods 
of  manufacturing  nitrogen  compounds  from  the  nitrogen  of  the  air. 

4.  Heating  Effects.  —  Fig.  383 
shows  the  plan  of  the  "electric  bomb." 
It  is  usually  made  of  wood.  Fill  the 
hole  with  gun  powder  as  far  up  as 
the  brass  rods  and  close  the  mouth 
with  a  wooden  ball.  Connect  the 
rods  to  the  poles  of  the  induction 
coil.  The  sparks  will  ignite  the 
powder  and  the  ball  will  be  projected 
across  the  room. 

The  heating  effect  of  the  current 
in  the  secondary  of  a  large  induction  {  1 
coil  may  be  shown  by  stretching  be- 
tween its  poles  a  very  thin  iron  wire. 
It  will  melt  and  burn  vividly.  If 
there  is  a  small  break  in  the  wire,  the  discharge  will  melt  the  part 
connected  to  the  negative  pole  of  the  coil,  while  the  other  part  will 
remain  below  the  temperature  of  ignition. 

470.   Discharges  in  Partial  Vacua.  —  Place  a  vase  of  uranium 

on  the  table  of  the  air  pump,  under  a  bell  jar  provided  with  a  brass 
sliding  rod  passing  air-tight  through  the  cap  at 
the  top  (Fig.  384).  Connect  the  rod  and  the  air 
pump  table  to  the  terminals  of  the  induction 
coil.  When  the  air  is  exhausted  a  beautiful  play 
of  light  will  fill  the  bell  jar.  The  display  will  be 
more  beautiful  if  the  vase  is  lined  part  way  up 
with  tin-foil.  This  experiment  is  known  as 
Gassiofs  cascade.  The  experiment  may  be  varied 
by  admitting  other  gases  and  exhausting  again. 
The  aspect  of  the  colored  light  will  be  entirely 
changed. 
Fig.  384 

The  best  effects  are  obtained  with  dis- 
charges from  the  secondary  of  an  induction  coil  in  glass 
tubes  when  the  exhaustion  is  carried  to  a  pressure  of 
about  2  mm.  of  mercury,  and  the  tubes  are  permanently 


364 


ELECTEOMAGNETIC  INDUCTION 


sealed.  Platinum  electrodes  are  melted  into  the  glass  at 
the  two  ends.  Such  tubes  are  known  as  Greissler  tubes. 

They  are  made  in  a  great 
variety  of  forms  (Fig.  385), 
and  the  luminous  effects 
are  more  intense  in  the  nar- 
row connecting  tubes  than 
in  the  large  bulbs  at  the 
ends.  The  colors  are  de- 
termined by  the  nature  of 
the  residual  gas.  Hydro- 
Fig*  385  gen  glows  with  a  brilliant 

crimson ;  the  vapor  of  water  gives  the  same  color,  indi- 
cating that  the  vapor  is  dissociated  by  the  discharge.  An 
examination  of  this  glow  by  the  spectroscope  gives  the 
characteristic  lines  of  the  gas  in  the  tube. 

Geissler  tubes  often  exhibit  stratifications,  which  consist 
of  portions  of  greater  brightness  separated  by  darker  in- 
tervals. Stratifications  have  been  produced  throughout  a 
tube  50  feet  long.  These  stratifications  or  striae  present 
an  unstable  flickering  motion,  re- 
sembling that  sometimes  observed 
during  auroral  displays. 

471.  The  Discharge  Intermittent. 
—  On  a  disk  of  white  cardboard  about 
20  cm.  in  diameter  paste  disks  of  black 
paper  2  cm.  in  diameter  (Fig.  386).  Ro- 
tate the  disk  rapidly  by  means  of  a 
whirling  table  or  an  electric  motor  and 
illuminate  it  by  a  Geissler  tube  in  a 
dark  room.  The  black  spots  will  be 
sharp  in  outline  because  each  flash  is  nearly  instantaneous  ;  while  the 
spots  in  the  different  circles  will  either  stand  still,  rotate  forward,  or 
rotate  backward.  If  in  the  brief  interval  between  the  flashes  the 


Fig.  386 


THE  INDUCTION   COIL 


365 


disk  rotates  through  an  angle  equal  to  that  between  the  spots  in  one 
of  the  circles,  the  spots  will  appear  to  stand  still ;  if  it  rotates  through 
a  slightly  greater  angle,  the  spots  will  appear  to  move  slowly  forward ; 
if  through  a  smaller  angle,  they  will  appear 
to  move  slowly  backward. 

Mount  a  Geissler  tube  on  a  frame  at- 
tached to  the  axle  of  a  small  electric  motor 
(Fig.  387).  Illuminate  the  tube  by  an  in- 
duction coil  while  it  rotates.  Star-shaped 
figures  will  be  seen,  consisting  of  a  number 
of  images  of  the  tube,  the  number  depend- 
ing on  the  speed  of  the  motor  as  compared 
with  the  period  of  vibration  of  the  circuit 
breaker. 

472.  Cathode  Rays.  —  When  the 
gas  pressure  in  a  tube  is  reduced  be- 
low about  a  millionth  of  an  atmos- 
phere, the  character  of  the  discharge 
is  much  altered.  The  positive  col- 
umn of  light  extending  out  from  the  anode  gradually 
disappears,  and  the  sides  of  the  tube  glow  with  brilliant 
phosphorescence.  With  English  glass  the  glow  is  blue; 
with  German  glass  it  is  a  soft  emerald.  The  luminosity 
of  the  glass  is  produced  by  a  radiation  in  straight  lines 
from  the  cathode  of  the  tube;  this  radiation  is  known 

as  cathode  rays.  They 
w^ere  first  studied  by 
Sir  William  Crookes, 
and  the  tubes  for  the  pur- 
pose are  called  Crookes 
tubes. 

Many  other  substances  be- 
sides glass  are  caused  to  glow 
by  the  impact  of  cathode  rays 
Fig.  388  (Fig.    388),   such    as    ruby, 


Fig.  387 


366 


ELECTROMAGNETIC  INDUCTION 


diamond,  and  various  sulphides.     The  color  of  the  glow  depends  on 
the  substance. 

Cathode  rays,  unlike  rays  of  light,  are  deflected  by  a  magnet,  and 


'        ' 

Fig.  389 

when    once  deflected    they  do    not  regain   their    former    direction 
(Fig.  389).     Cathode  rays  proceed  in  straight  lines,  except  as  they 

are  deflected  by  a  magnet  or  by 
mutual  repulsion.  A  screen  placed 
across  their  path  interrupts  them  and 
casts  a  shadow  on  the  walls  of  the  tube. 
When  the  cathode  is  made  in  the 
form  of  a  concave  cup,  the  rays  are 
brought  to  a  focus  at  its  center  of 
curvature;  platinum  foil  placed  at 
this  focus  is  raised  to  bright  incan- 
descence and  may  be  fused  (Fig. 
390).  Glass  on  which  an  energetic 
cathode  stream  falls  may  be  heated 
to  the  point  of  fusion. 

It  has  been  conclusively 
shown  that  cathode  rays  carry 
negative  charges  of  electricity. 
Hence  the  mutual  repulsion 
exerted  on  each  other  by  two 
Fig.  390  parallel  cathode  streams. 

473.   Roentgen  Rays.  —  The  rays  of  radiant  matter,  as 
Crookes  called  it,  emanating  from  the  cathode,  give  rise  to 


Sir  William  Crookes,  a  dis- 
tinguished English  chemist, 
was  born  in  1832.  In  1873 
he  began  a  series  of  investi- 
gations on  the  properties  of 
high  vacua.  While  engaged 
in  this  work  he  invented  the 
radiometer,  developed  the 
Crookes  tubes,  and  dis- 
covered what  he  called  "ra- 
diant matter."  His  investi- 
gations led  him  very  close  to 
the  discoveries  of  Rontgen. 
He  has  edited  the  Quarterly 
Journal  of  Science  since 
1864. 


Wilhelm  Konrad  Rontgen 

was  born  in  1845.  It  was 
at  Wiirzburg,  Germany,  in 
1895,  that  he  discovered 
while  passing  electric  charges 
through  a  Crookes  tube,  that 
a  certain  kind  of  radiation 
was  emitted  capable  of  pass- 
ing through  many  substances 
known  to  be  opaque  to  light. 
The  nature  of  these  rays 
being  unknown,  he  called 
them  "X-rays."  They  differ 
from  the  cathode  rays  dis- 
covered by  Crookes,  in  that 
they  affect  a  sensitized  photo- 
graphic plate. 


THE  INDUCTION  COIL  367 

another  kind  of  rays  when  they  strike  the  walls  of  the 
tube,  or  a  piece  of  platinum  placed  in  their  path.  These 
last  rays,  to  which  Roentgen,  their  discoverer,  gave  the 
name  of  "X-rays"  can  pass  through  glass,  and  so  get 
out  of  the  tube.  They  also  pass  through  wood,  paper, 
flesh,  and  many  other  substances  opaque  to  light.  They 
are  stopped  by  bones,  metals  (except  in  very  thin  sheets), 
and  by  some  other  substances.  Roentgen  discovered  that 
they  affect  a  photographic  plate  like  light.  Hence,  photo- 
graphs can  be  taken  of  objects  which  are  entirely  invisible 
to  the  eye,  such  as 
the  bones  in  a  liv- 
ing body,  or  bul- 
lets embedded  in 
the  flesh. 

A  Crookes  tube 
adapted  to  the  pro- 
duction of  Roent-  Fig.  391 
gen  rays  (Fig.  391)  has  a  concave  cathode  K,  and  at  its 
focus  an  inclined  piece  of  platinum  A,  which  serves  as  the 
anode.     The  X-rays  originate  at  A  and  issue  from  the  side 
of  the  tube. 

474.  X-Ray  Pictures.  — The  penetrating  power  of  Roentgen  rays 
depends  largely  on  the  pressure  within  the  tube.  With  high  exhaus- 
tion the  rays  have  high  penetrating  power  and  are  then  known  as 
"  hard  rays."  Hard  rays  can  readily  penetrate  several  centimeters  of 
wood,  and  even  a  few  millimeters  of  lead.  With  somewhat  lower 
exhaustion,  the  rays  are  less  penetrating  and  are  then  known  as  "soft 
rays." 

The  possibility  of  X-ray  photographs  depends  on  the  variation  in 
the  penetrability  of  different  substances  for  X-rays.  Thus,  the  bones 
of  the  body  absorb  Roentgen  rays  more  than  the  flesh,  or  are  less  pene- 
trable by  them.  Hence  fewer  rays  traverse  them.  Since  Roentgen 
rays  cannot  be  focused,  all  photographs  taken  by  them  are  only  shadow 


368 


ELECTEOMAGNETIC  INDUCTION 


pictures.  A  Roentgen 
photograph  of  a  gloved 
hand  is  shown  in  Fig.  392. 
The  ring  on  the  little  fin- 
ger, the  two  glove  buttons, 
and  the  cuff  studs  are 
conspicuous.  The  flesh  is 
scarcely  visible  because  of 
the  high  penetrating  power 
of  the  rays  used.  The 
photographic  plate  for  the 
purpose  is  inclosed  in  an 
ordinary  plate  holder  and 
the  hand  is  laid  on  the 
holder  next  to  the  sensi- 
tized side. 

475.  The  Fluore- 
scope.  —  Soon  after  the 
discovery  of  X-rays 
it  was  found  that  cer- 
tain fluorescent  sub- 


Fig.  392 

stances,  like  platino-barium- cyanide,  and  calcium  tungstate, 
become  luminous  under  the  action  of  X-rays.  This  fact 
has  been  turned  to  account 
in  the  construction  of  a 
fluorescope  (Fig.  393),  by 
means  of  which  shadow  pic- 
tures of  concealed  objects 
become  visible.  An  opaque 
screen  is  covered  on  one  side 
with  the  fluorescent  sub- 

p- 

stance  ;  this  screen  fits  into 
the  larger  end  of  a  box  blackened  inside,  and  having  at 
the  other  end  an  opening  adapted  to  fit  closely  around 
the  eyes,  so  as  to  exclude  all  outside  light.  When  an  ob- 


THE  INDUCTION   COIL  369 

ject,  such  as  the  hand,  is  held  against  the  fluorescent 
screen  and  the  fluorescope  is  turned  toward  the  Roentgen 
tube,  the  bones  are  plainly  visible  as  darker  objects  than 
the  flesh  because  they  are  more  opaque  to  X-rays.  The 
beating  heart  may  be  made  visible  in  a  similar  manner. 

476.  Radioactivity.  —  Henri    Becquerel    of    Paris   dis- 
covered in  1896  that  a  double  sulphate  of  potassium  and 
uranium  emits  radiations  which  affect  a  photographic  plate 
in  the  same  way  as  the  X-rays.     All  substances  which 
emit  radiations  of  this  character  are  said  to  be  radioactive. 
The  principle  ones  are  compounds  of  uranium,  polonium, 
actium,  and  thorium. 

In  1898,  M.  and  Mme.  Curie  by  chemical  methods  sepa- 
rated from  pitchblende,  an  ore  containing  uranium,  three 
substances  each  more  highly  radioactive  than  uranium  ; 
these  were  polonium,  actium,  and  radium.  Pure  radium 
has  not  been  obtained  ;  it  is  used  in  investigations  in  the 
form  of  a  chloride  or  bromide,  and  in  that  form  its  radio- 
activity is  more  than  a  million  times  greater  than  that  of 
the  pitchblende  from  which  it  is  derived.  Its  emanations 
excite  strong  fluorescence  in  some  substances;  their  action 
on  the  human  body  is  to  produce  sores  difficult  to  heal. 
Radium  is  a  very  unstable  substance,  tending  to  disinte- 
grate into  other  things.  During  these  changes  large 
quantities  of  energy  in  the  form  of  heat  are  given  off,  a 
gram  of  radium  yielding  100  calories  of  heat  per  hour. 
It  has  been  calculated  that  this  emission  of  energy  will 
continue  for  a  period  of  over  2600  years  before  exhaustion 
is  reached. 

477.  The  Electron  Theory  of  Matter.  —  Facts  revealed  in 
the  study  of  electric  charges  through  high  vacuum  tubes 
by  Crookes,  J.  J.   Thomson,  and  others,  along  with  the 


370  ELECTROMAGNETIC  INDUCTION 

revelations  of  radioactive  substances,  make  it  certain 
that  the  atom  of  chemistry  is  a  compound  and  very  com- 
plex in  structure.  By  methods  too  complex  to  describe 
here,  Thomson  has  seemingly  shown  that  the  ultimates  of 
matter  are  minute  particles,  variously  called  corpuscles  or 
electrons.  These  carry  negative  charges  of  electricity, 
and  revolve  about  one  another  in  very  intricate  orbits,  the 
number  of  electrons  and  the  character  of  their  motions 
determining  the  nature  of  the  atom  which  they  compose, 
whether  it  is  one  of  gold,  lead,  hydrogen,  or  what  not. 

Each   electron   is    calculated   to   be  part  of  a  centi- 


meter in  diameter,  700  of  them  making  up  an  atom  of 
hydrogen,  15,000  an  atom  of  sodium,  100,000  an  atom  of 
mercury,  and  160,000  an  atom  of  radium,  the  weight  of  the 
atom  being  governed  by  the  number  of  electrons  compos- 
ing it.  The  empty  space  in  an  atom  is  1010  times  greater 
than  the  space  filled  by  the  electron.  There  remains  these 
mysteries  to  be  solved  :  What  are  electrons  ?  What  does 
it  mean  to  say  that  they  are  negative  electric  charges  ? 
What  are  positive  electric  charges  ?  What  is  electricity  ? 
Can  we  hope  ever  to  control  the  groupings  of  these  elec- 
trons and  produce  any  kind  of  matter  at  will,  thereby 
realizing  the  dream  of  the  ancient  alchemists  ?  Will  it 
ever  be  possible  to  release  the  vast  stores  of  energy  locked 
up  in  the  atom  for  the  use  of  mankind  when  other  stores 
of  energy  have  been  exhausted  ? 


Madame  Marie  Sklodowska  Curie  was  born  in  Warsaw  in 
1867.  She  imbibed  the  spirit  of  scientific  research  from  her 
father,  a  distinguished  physicist  and  chemist.  In  1895  she  mar- 
ried Professor  Curie  of  the  University  of  Paris.  Three  times  she 
has  been  awarded  the  Gegner  prize  by  the  French  Academy  for 
her  valuable  contributions  to  the  world's  knowledge  of  the  mag- 
netic properties  of  iron  and  steel  and  for  her  discoveries  in  radio- 
activity. In  1903  and  again  in  1911  the  Nobel  prize  was  awarded 
her.  In  January  of  191 1  she  failed  only  by  two  votes  of  election 
to  membership  in  the  French  Academy  of  Sciences,  being  de- 
feated by  Branley,  the  inventor  of  the  coherer  used  in  the  Mar- 
coni system  of  wireless  telegraphy. 


CHAPTER   XIV 
DYNAMO-ELECTRIC  MACHINERY 
I.  DIRECT  CURRENT  MACHINES 

478.  A   Dynamo -Electric    Machine   converts    mechanical 
energy  into  the  energy  of  currents  of  electricity.     It  is  a 
direct   outgrowth   of   the    discoveries   of   Faraday  about 
induced  electromotive  forces  and  currents  in  1881.     It  is 
an  essential  part  of  every  system,  steam  or  hydroelectric, 
for  electric  lighting,  the  transmission  of  electric  power, 
electric    railways,    electric    locomotives,    electric    smelt- 
ing, electrolytic  refinement  of  metals,  electric  train  light- 
ing, the  charging  of  storage  batteries,  and  for  every  other 
use  to  which  large  electric  currents  are  applied. 

Every  dynamo-electric  machine  has  three  essential 
parts  :  1.  The  field  magnet  to  produce  a  powerful  mag- 
netic field.  2.  The  armature,  a  system  of  conductors 
wound  on  an  iron  core,  and  revolving  in  the  magnetic 
field  in  such  a  manner  that  the  magnetic  flux  through 
these  conductors  varies  continuously.  3.  The  commutator, 
or  the  collecting  rings  and  the  brushes,  by  means  of  which 
the  machine  is  connected  to  the  external  circuit.  If  the 
magnetic  field  is  produced  by  a  permanent  magnet,  the 
machine  is  called  a  magneto ;  if  by  an  electromagnet, 
the  machine  is  a  dynamo.  They  are  both  often  called 
generators. 

479.  Ideal  Simple  Dynamo. —  Suppose   a   single   loop   of 
wire  to  revolve  between  the  poles  of  a  magnet  (Fig.  394) 

371 


372  DYNAMO-ELECTRIC  MACHINERY 

in  the  direction  of  the  arrow  and  around  a  horizontal  axis. 
The  light  lines  indicate  the  magnet  flux  running  across 
from  JV  to  jS.  In  the  position  of  the  loop  drawn  in  full 

lines  it  incloses  the 
largest  possible  magnetic 
flux  or  lines  of  force,  but 
as  the  flux  inclosed  by 
the  coil  is  not  changing, 
the  induced  E.  M.  F.  is 
394  zero.  When  it  has  ro- 

tated forward  a  quarter 
of  a  turn,  its  plane  will  be  parallel  to  the  magnetic  flux, 
and  no  lines  of  force  will  then  pass  through  it.  During 
this  quarter  turn  the  decrease  in  the  magnetic  flux 
threading  through  the  loop  generates  a  direct  E.  M.  F. ; 
and  if  the  rotation  is  uniform,  the  rate  of  decrease  of  flux 
through  the  loop  increases  all  the  way  from  the  first 
position  to  the  one  shown  by  the  dotted  lines,  where  it  is 
a  maximum.  The  arrows  on  the  loop  show  the  direction 
of  the  E.  M.  F.  During  the  next  quarter  turn  there  is  an 
increase  of  flux  through  the  loop,  but  it  runs  through  the 
loop  in  the  opposite  direction  because  the  loop  has  turned 
over ;  this  is  equivalent  to  a  continuous  decrease  in  the 
original  direction,  and  therefore  the  direction  of  the  induced 
E.  M.  F.  around  the  loop  remains  the  same  for  the  entire 
half  turn,  and  the  E.  M.  F.  again  becomes  zero  when  the 
half  turn  is  completed.  After  the  half  turn,  the  conditions 
are  all  reversed  and  the  E.  M.  F.  is  directed  the  other  way 
around  the  loop.  If  the  loop  is  part  of  a  closed  circuit, 
the  current  through  it  reverses  twice  every  revolution. 

480.    The  Commutator.  —  When  it  is  desired  to  convert 
the  alternating  currents  flowing  in  the  armature  into  a 


DIRECT  CURRENT  MACHINES 


373 


Fig.  395 


current  in  one  direction  through  the  external  circuit,  a 
special  devic6  called  a  commutator  is  emploj'ed.  For  a  sin- 
gle coil  in  the  armature,  the  commutator  consists  of  two 
parts  only.  It  is  a  split  tube  with  the  two  halves,  a  and  5, 
insulated  from  each  other  and  from  the  shaft  S  on  which 
they  are  mounted  (Fig.  395). 
The  two  ends  of  the  coil  are 
connected  with  the  two  halves 
of  the  tube.  Two  brushes, 

with  which  the  external  cir- 

•.  •  •  /*  -J  JF' 

cuit  LLis  conne-ctecMffear  on 

the  commutator,  and  they  are 
so  placed  that  they  exchange 
contact  with  the  two  commutator  segments  at  the  same 
time  that  the  current  reverses  in  the  coil.  In  this  wtiy 
one  of  the  brushes  is  always  positive  and  the  other  nega- 
tive, and  the  current  flows  in  the  external  circuit  from  the 
positive  brush  back  to  the  negative,  and  thence  through 
the  armature  to  the  positive  again. 

481.  The  Gramme  Ring.  —  The  use  of  a  commutator  with 
more  than  two  parts  is  conveniently  illustrated  in  con- 
nection with  the  Gramme 
ring.  This  armature  has  a 
core  made  either  of  iron  wire, 
or  of  thin  disks  at  right  angles 
to  the  axis  of  rotation.  The 
iron  is  divided  for  the  pur- 
pose of  preventing  induction 
currents  in  it,  which  waste 


Fig.  396 


energy.  The  relation  of  the  several  parts  of  the  machine 
is  illustrated  by  Fig.  396.  A  number  of  coils  are  wound 
in  one  direction  and  are  all  joined  in  series.  Each  junction 


374 


D  YNAMO-ELECTRIC  MA  CHINEE  Y 


between  coils  is  connected  with  a  commutator  bar.  Most 
of  the  magnetic  flux  passes  through  the  iron  ring  from  the 
north  pole  side  to  the  south  pole  ;  hence,  when  a  coil  is 
in  the  highest  position  in  the  figure,  the  maximum  flux 
passes  through  it;  as  the  ring  rotates,  the  flux  through  the 
coil  decreases,  and  after  a  quarter  of  a  revolution  there  is 
no  flux  through  it.  The  current  through  each  coil  reverses 
twice  during  each  revolution,  exactly  as  in  the  case  of  the 
single  loop.  No  current  flows  entirely  around  the  arma- 
ture, because  the  E.  M.  F.  generated  in  one  coil  at  any  in- 
stant is  exactly  counterbalanced  by  the  E.  M.  F.  generated  in 
the  coil  opposite.  But  when  the  external  circuit  connecting 
the  brushes  is  closed,  a  current  flows  up  on  both  sides  of 
the  armature.  The  current  has  then  two  paths  through 
ttfe  armature,  and  one  brush  is  constantly  positive  and  the 
other  negative. 

482.    The  Field  Magnet.  —  The  magnetic  field  in  dynamos 
is  produced  by  a  large  electromagnet  excited  by  the  cur- 


Main  Circuit  J 

Fig.  397 


Fig.  398 


rent  flowing  from  the  armature  ;  this  current  is  led,  either 
wholly  or  in  part,  around  the  field-magnet  cores.    When  the 


DIRECT  CURRENT  MACHINES 


375 


entire  current  is  carried  around  the  coils  of  the  field  mag- 
net, the  dynamo  is  said  to  be  series  wound  (Fig.  397). 
When  the  field  magnet  is  excited 
by  coils  of  many  turns  of  fine  wire 
connected  as  a  shunt  to  the  ex- 
ternal circuit,  the  dynamo  is  said 
to  be  shunt  wound  (Fig.  398).  A 
combination  of  these  two  methods 
of  exciting  the  field  magnet  is 
called  compound  winding  (Fig. 
399).  The  residual  magnetism  re- 
maining in  the  cores  is  sufficient 
to  start  the  machine.  The  cur- 
rent thus  produced  increases  the 


Fig.  399 


magnetic  flux  through  the  armature  and  so  increases  the 
E.  M.  F. 

483.    The  Drum  Armature. — This  very  useful  form  of 
armature  is  shown  at  A  in  Fig.  400.     It  consists  of  an 


Fig.  400 


3T6 


DYNAMO-ELECTEIC  MACHINERY 


iron  core  of  laminated  disks,  in  which  are  cut  a  series  of 
grooves  parallel  to  the  shaft,  and  coils  wound  in  them  at 
equal  angular  distances  around  the  circumference.  (The 
bands  shown  in  the  figure  serve  only  to  keep  the  coils  in 
place.)  All  the  coils  of  the  armature  may  be  joined  in 
series,  and  the  junctions  between  them  are  then  connected 
to  the  commutator  bars  (7,  as  in  the  Gramme  ring. 

When  the  number  of  coils  is  twenty  or  more,  the  poten- 
tial difference  between  the  brushes  never  drops  to  zero,  as 

it  does  in  the  case 
of  a  single  coil 
(§479),  but  it  re- 
mains nearly  con- 
stant. To  reduce 
the  rate  of  rota- 
tion of  the  arma- 
ture,field  magnets 
of  four,  six,  eight, 
or  more  poles  are 
used.  In  the  ma- 
chine shown  in 
Fig.  400  there 
are  four  poles 
and  four  sets  of 
brushes.  Two  of 


Fig;  401 


the  brushes  are  positive  and  two  negative ;  the  two  posi- 
tive brushes  are  connected  in  parallel  to  form  one  positive, 
and  the  same  is  true  of  the  two  negative  ones.  Fig.  401 
is  the  assembled  machine  ready  to  run. 

484.  The  Electric  Motor.  —  The  electric  motor  is  a  machine 
for  the  reconversion  of  the  energy  of  electric  currents  into 
mechanical  power. 


DIRECT  CURRENT  MACHINES 


377 


In  the  electric  automobile  the  motor  is  driven  by  currents  from  a 
storage  battery.  In  the  electric  street  car  it  derives  its  current  and 
power  from  a  trolley,  a  third  rail,  or  from  conductors  fixed  in  a  slotted 
conduit  under  the  pavement,  all  of  them  leading  back  to  a  power 
house  or  a  substation.  The  electric  motor  is  extensively  used  for 
small  power  as  well  as  for  large  units.  Witness  the  use  of  electric 
fans,  electric  coffee  grinders,  sewing  machine  motors,  and  electrically 
driven  bellows  for  pipe  organs  on  one  hand,  and  on  the  other  the  elec- 
tric drive  for  large  fans  to  ventilate  mines  and  buildings,  electric 
elevators,  and  electrically  driven  mills  and  factories. 

An  electric  motor  for  direct  currents  is  constructed  in 
the  same  manner  as  a  generator.  In  fact/ any  direct 
current  generator  may  be  used  as  a  motor.  A  study  of 
the  magnetic  field  resulting  from  the  interaction  of  the 
field  of  the  field  magnet  and  that  of  a  single  loop  carrying 
an  electric  current  will  make  it  clear  that  such  a  loop  has 
a  tendency  to  rotate. 
In  Fig.  402  the  field 
between  unlike  poles 
is  distorted  by  a  cur- 
rent through  a  loop 
of  wire,  which  came 
up  through  one  of 
the  holes  shown  and 
went  down  through 
the  other.  These 
lines  of  force  are  Fi  4Q2 

under    tension    and 

tend  to  straighten  out ;  there  is  therefore  a  magnetic 
stress  acting  on  the  loop  and  tending  to  turn  it  counter- 
clockwise. If  the  loop  is  allowed  to  rotate  in  the  direc- 
tion of  this  magnetic  effort  between  the  field  and  the  loop, 
the  loop  is  an  armature  and  work  is  done  by  the  machine 
as  a  motor.  But  if  the  loop  is  rotated  clockwise  by  me- 


3T8 


DYNAMO-ELECTRIC  MACHINERY 


chanical  means,  it  turns  against  the  magnetic  effort  acting 
on  it,  and  work  must  be  done  against  the  resistance  of 
this  magnetic  drag.  The  loop  is  then  the  armature  of  a 
generator. 

II.    ALTERNATORS  AND  TRANSFORMERS 

485.   The  Alternator.  —  If   the   brushes  A   and   B   of   a 

dynamo  bear  on  two  continuous  rings  mounted  on  the 

shaft  -(Fig.  403),  instead  of  on  a 
commutator,  the  current  in  the 
external  circuit  WW  will  alter- 
nate or  reverse,  as  it  does  in  an 
armature  coil,  every  time  the  ar- 
mature turns  through  the  angular 
distance  from  one  pole  to  the  next. 

A  complete  series  of  changes  in  the  current  and  E.  M.  F. 

in   both   directions  takes   place    while    the    armature   is 

turning  from  one  pole 

to  the  next  one  of  the 

same   name.      Such  a 

series    of    changes    is 

called    a    cycle.      The 

frequency  is  equal  to 

the    product    of     the 

number    of     pairs    of 

poles  on  the  field  mag- 
net and  the  number  of 

rotations  per    second. 

Frequencies   are    now 

restricted  between  the  p.     4Q4 

limits  of  about  25  and 

60  cycles  per  second.     Multipolar  machines  are  used  to 

avoid  excessive  speed  of  rotation. 


ALTERNATORS  AND   TRANSFORMERS  379 

Figure  404  is  a  diagram  of  an  alternator  with  a  stationary 
field  outside  and  an  armature  rotating  with  the  shaft. 
The  field  is  excited  by  a  direct  current  machine.  The 
armature  coils  are  reversed  in  winding  from  one  field  pole 
N  to  the  next  $,  they  are  joined  in  series,  and  the  termi- 
nals are  brought  out  to  rings  on  the  shaft.  The  brushes 
bearing  on  these  rings  lead  to  the  external  circuit. 

486.  Transformers.  —  A  transformer  is  an  induction  coil 
witli  a  primary  of  many  turns  of  wire  and  a  secondary  of 
a  smaller  number,  both  wound  around  a  divided  iron  core 
forming  a  closed  magnetic  circuit; 
that  is,  one  magnetic  circuit  is  inter- 
linked with  two  electric  circuits 
(Fig.  405).  A  transformer  is  em- 
ployed with  alternating  currents 
either  to  step  down  from  a  high 
E.  M.  F.  to  a  low  one,  or  the  re- 
verse. The  two  electromotive  forces 
are  directly  proportional  to  the  num- 
ber of  turns  of  wire  in  the  two  coils. 
For  example,  to  reduce  a  2000-volt 

current  to  a  100-volt  current,  there  must  be  20  turns  in  the 
primary  to  every  one  in  the  secondary.  Both  coils  are 
wound  on  the  same  iron  core,  and  are  as  perfectly  insulated 
from  each  other  as  possible.  The  iron  serves  as  a  path 
for  the  flux  of  magnetic  induction,  and  all  the  lines  of 
force  produced  by  either  coil  pass  through  the  other,  ex- 
cept for  a  small  amount  of  "magnetic  leakage."  When 
the  secondary  is  open,  the  transformer  acts  simply  as  a 
"choke  coil";  that  is,  the  self-induction  of  the  primary  is 
so  large  that  only  sufficient  current  is  transmitted  to 
magnetize  the  iron  and  to  furnish  the  small  amount  of 


380  DYNAMO-ELECTRIC  MACHINERY 

energy  lost  in  it.  The  counter-E.  M.  F.  of  self-induction 
is  then  nearly  equal  to  the  E.  M.  F.  impressed  from  with- 
out. But  when  the  secondary  is  closed,  the  self-induction 
is  suppressed  to  the  extent  that  the  transformer  auto- 
matically adjusts  itself  to  the  condition  that  the  energy 
in  the  secondary  circuit  lacks  only  a  few  per  cent  of  the 
energy  absorbed  by  the  primary  from  the  generator. 

487.  Transformers  in  a  Long-distance  Circuit.  —  The  util- 
ity of  the  transformer  lies  in  its  use  to  secure  high  voltage 
for  transmission  and  low  voltage  for  lighting  and  power. 
Only  small  currents  can  be  transmitted  over  distances 
exceeding  a  few  hundred  feet  without  excessive  heat  losses 
on  account  of  the  resistance  of  the  conductors.  To  transmit 
power  while  still  keeping  the  current  small,  the  electric 
pressure,  that  is,  the  number  of  volts,  must  be  increased, 
for  power  transmitted  in  watts  is  proportional  to  the  prod- 
uct of  the  number  of  volts  and  the  number  of  amperes. 
Transformers  are  in  actual  use  on  long-distance  circuits 
for  raising  the  voltage  to  100,000  or  more  volts  potential 
difference  between  the  main  long-distance  wires.  Fig. 
406  is  a  diagram  showing  a  transformer  system  for  long- 


-oH 


distance  power  transmission.  The  first  transformer  A. 
raises  the  potential  difference  from  2000  volts  to  50,000 
volts.  The  long  distance  transmission  takes  place  at  this 
voltage  to  the  second  transformer  B,  which  steps  down 
from  50,000  to  2000  volts  for  local  transmission  within 


Sir  Joseph  John  Thomson  was  born  near  Manchester,  Eng- 
land, in  1856.  He  received  his  early  training  at  Owens  College, 
and  acquired  there  some  knowledge  of  experimental  work  in  the 
laboratory  of  Balfour  Stewart.  At  the  age  of  twenty-seven  he 
was  appointed  to  the  Cavendish  professorship  at  the  University  of 
Cambridge,  a  position  made  famous  by  Maxwell  and  Rayleigh. 
The  wisdom  of  the  appointment  was  soon  proved ;  for  shortly 
after,  Thomson  began  a  series  of  experiments  on  the  conduction 
of  electricity  through  gases,  culminating  in  the  discovery  of  the 
"electron,"  out  of  which  has  Developed  the  electron  theory  of 
matter. 


ELECTRIC  LIGHTING 


381 


the  limits  of  a  city.  The  third  transformer  (7  steps  down 
further  from  2000  to  100  volts  for  house  service  for  light- 
ing, fan  motors,  electric  cooking,  electric  flat  irons,  etc. 

III.     ELECTRIC  LIGHTING 

488.  The  Carbon  Arc.  —  In  1800  Sir  Humphry  Davy 
discovered  that  when  two  pieces  of  charcoal  suitably  con- 
nected to  a  powerful  voltaic  battery  were  brought  into 
contact  at  their  ends  and  were  then  separated  a  slight  dis- 
tance, brilliant  sparks  passed  between  them.  No  mention 
was  made  of  the  electric  arc  until  1808.  With  a  battery 
of  2000  cells  and  the  carbons  in  a  horizontal  line,  they 
could  be  separated  several  inches,  while  the  current  was 
conducted  across  in  the  form  of  a  curved  flame  or  arc. 
Hence  the  name  electric  arc  given 
to  this  form  of  electric  lighting. 

Dense  compressed  or  molded 
carbon  rods  are  now  used,  and 
when  they  are  separated  a  slight 
distance  they  are  heated  to  an 
exceedingly  high  temperature, 
and  the  current  from  a  dynamo 
continues  to  pass  across  through 
the  heated  carbon  vapor.  The 
ends  of  the  carbon  rods  in  the 
open  air  are  disintegrated,  a  de- 
pression or  "crater"  forming 
in  the  positive  and  a  cone  on  the 
negative  (Fig.  407).  Most  of 
the  light  of  the  open  arc  comes 

from  the  bottom  of  this  crater,  the  temperature  of  which 
Violle  has  estimated  to  be  3500°  C.  The  arc  light  may 
be  produced  in  a  vacuum.  The  intense  heat  is  not,  there- 


Fig.  407 


382 


DYNAMO-ELECTRIC  MACHINERY 


fore,  generated  by  combustion.     It  is  the  energy  of  the 
current  converted  into  heat  by  the  resistance  of  the  arc. 

489.  The  Open  and  the  Inclosed  Arc.  —  To  keep  the  carbon 
rods  from  burning  away  too  rapidly,  modern  arc  lamps  are 

mostly  of  the  "  inclosed  arc  "  type.  The  lower 
carbon  and  a  part  of  the  upper  one  are  in- 
closed in  a  small  glass  globe,  which  is  air-tight 
at  the  bottom,  but  allows  the  upper  carbon  to 
slip  through  a  check- valve  at  the  top  (Fig.  408). 
Soon  after  the  arc  begins  to  burn,  the  oxygen 
in  the  globe  is  absorbed  and  the  arc  is  then 
inclosed  in  an  atmosphere  of  nitrogen  from  the 
air  and  of  carbon  monoxide  from  the  incomplete 
combustion  of  the  carbon.  The  inclosed  arc  is 
longer  than  the  open  arc,  and  the  E.  M.  F.  is 
about  80  volts  instead  of  50  as  required  by  the 
open  arc;  but  the  current  for  the  inclosed  arc  is  smaller 
than  for  the  open  arc.  The  carbons  for  the  inclosed  arc 
last  about  ten  times  as  long  as  in  the  open  air. 

490.  Other  Arc  Lights.  —  Other  arc  lamps  are  now  in 
commercial  use  in  which  the  light  comes  chiefly  from  the 
incandescent  stream  between  the  electrodes.     They  have 
a  higher  efficiency  than  the  carbon  arc.     In  the  "metallic 
arc  "  powdered  magnetite  in  an  iron  tube  is  used  for  one 
electrode  and  a  block  of  copper  for  the  other.     The  arc 
flame  is  very  white  and  brilliant,  the  light  coming  from 
the  luminous  iron  vapor. 

"  Flaming  arcs  "  are  made  by  the  use  of  a  positive  elec- 
trode impregnated  with  salts  of  calcium.  The  light  from 
the  flaming  arc  is  yellow,  and  is  adapted  to  outdoor  illu- 
mination only. 


Fig.  408 


ELECTRIC  LIGHTING 


383 


Fig.  409 


491.  The  Incandescent  Lamp.  — The  heat  and  light  in  an 
incandescent  lamp  are  due  to  the  simple  resistance  of  a 
conducting  filament  inclosed  in  an  ex- 
hausted glass  globe  (Fig.  409).  The 
ends  of  the  filament  are  connected 
through  the  glass  by  means  of  short 
pieces  of  platinum  wire.  Platinum  is 
used  because  its  coefficient  of  expan- 
sion is  about  the  same  as  that  of  glass ; 
and  so,  when  the  lamp  becomes  hot  in 
use,  it  neither  leaks  air  around  the 
wires  nor  cracks. 

The   carbon   filament  is  usually  made   from    cellulose 
obtained  from  cotton.     The  temperature  to  which  a  car- 
bon filament  can  be  raised  is  limited  by  the  tendency  of 
the  carbon  to  disintegrate  at  high  tempera- 
tures.   The  carbon  thrown  off  rapidly  reduces 
the   thickness  of  the  filament  and  blackens 
the  globe.     The  useful  life  of  a  carbon  fila- 
ment is  from  600  to  800  hours. 

In  recent  years  filaments  have  been  made 
of  the  rare  metals,  osmium,  tantalum,  and 
tungsten.  The  tungsten  lamp  (Fig.  410)  is 
rapidly  displacing  the  carbon  lamp  because 
of  its  higher  efficiency,  in  spite  of  the  fact 
that  it  is  much  more  fragile. 

The  ordinary  commercial  unit  for  the  carbon  filament  is 
the  16-candle  power  incandescent  lamp.  On  a  110-volt 
circuit  it  takes  about  0.5  ampere.  Since  the  power  in 
watts  consumed  is  El,  this  lamp  consumes  about  55  watts, 
or  3.5  watts  per  candle  power.  The  tungsten  25- watt 
lamp  gives  20  candle  power,  and  the  40-watt  lamp  32  cau- 
dle power,  or  1.25  watts  per  candle. 


Fig.  410 


384 


D  YNA MO-ELECTRIC  MA  CHINER  Y 


492.  Incandescent  Lamp  Circuits.  —  Incandescent  lamps 
are  connected  in  parallel  between  the  mains  in  a  building. 
These  mains  lead  either  directly  to  a  dynamo  (Fig.  411), 


0  j 

>s    O     ( 

H 

i 
q 

•oJ 

Pri. 


Fig.  411 


Fig.  412 


or  to  the  low  voltage  side  of  a  transformer  in  the  case  of 
alternating  currents  (Fig.  412).  Single  lamps  are  turned 
off  usually  by  the  key  in  the  socket  (Fig.  409),  and 
groups  of  lamps  by  a  switch  8. 


IV.    THE  ELECTRIC  TELEGRAPH 

493.  The  Electric  Telegraph  is  a  system  of  transmitting 
messages  by  means  of  simple  signals  through  the  agency 
of  an  electric  current.     Its  essential  parts  are  the  line,  the 
transmitter  or  key,  the  receiver  or  sounder,  and  the  lattery. 

494.  The   Line   is  an  iron,   copper  or  phosphor-bronze 
wire,  insulated  from  the  earth  except  at  its  ends,  and 
serving  to  connect  the  signaling  apparatus.     The  ends  of 
this  conductor  are   connected  with  large  metallic   plates, 
or  with  gas  or  water  pipes,  buried  in  the  earth.     By  this 

means  the  earth  becomes  a  part 
of  the  electric  circuit  containing 
the  signaling  apparatus. 


495.    The    Transmitter    or   Key 
(Fig.  413)  is  merely  a  current 


Fig.  413 


THE  ELECTRIC   TELEGRAPH 


385 


Fig.  414 


interrupter,  and  usually  consists  of  a  brass  lever  A,  turn- 
ing about  pivots  at  B.  It  is  connected  with  the  line  by 
the  screws  O  and  D.  When  the  lever  is  pressed  down, 
a  platinum  point  projecting  under  the  lever  is  brought  in 
contact  with  another  platinum  point  J3,  thus  closing  the 
circuit.  When  not  in  use,  the  circuit  is  left  closed,  the 
switch  F  being  used  for 
that  purpose. 

496.  The  Receiver  or 
Sounder  (Fig.  414)  con- 
sists of  an  electromag- 
net  A  with   a  pivoted 
armature  B.     When  the 
circuit  is  closed  through 

the  terminals  D  and  E,  the  armature  is  attracted  to  the 
magnet,  producing  a  sharp  click.  When  the  circuit  is 
broken,  a  spring  0  causes  the  lever  to  rise  and  strike  the 
back  stop  with  a  lighter  click. 

497.  The  Relay.  —  When  the  resistance  of   the  line  is 
large,  the  current  is  not  likely  to  be  strong  enough  to  oper- 
ate the  sounder 
with     sufficient 
energy  to    ren- 
der the   signals 

t^^^_  '^^•**Ht          distinctly  audi- 

^  -  "r^K       ble.    To  remedy 

this  defect,  an 
electromagnet, 
called  a  relay  (Fig.  415),  whose  helix  A  is  composed  of 
many  turns  of  fine  wire,  is  placed  in  the  circuit  by  means 
of  its  terminals  Q  and  D.  As  its  armature  moves  to  and 


Fig.  415 


886 


DYNAMO-ELECTRIC  MACHINERY 


fro  between  the  points  at  K^  it  opens  and  closes  a  second 
and  shorter  circuit  through  j^and  F,  in  which  the  sounder 
is  placed.  Thus  the  weak  current,  through  the  agency  of 
the  relay,  brings  into  action  a  current  strong  enough  to  do 
the  necessary  work. 

498.  The  Battery  consists  of  a  large  number   of   cells, 
usually  of  the  gravity  type,   connected  in  series.     It  is 
generally  divided  into  two  sections,  one  placed  at  each 
terminal  station,  these  sections  being  connected  in  series 
through   the  line.     The  principal    circuits  of  the   great 
telegraph  companies  are  now  worked  by  means  of  currents 
from  dynamo  machines. 

499.  The  Signals  are  a  series  of  sharp  and  light  clicks 
separated  by  intervals  of  silence  of  greater  or  less  dura- 
tion, a  short  interval  between  the  clicks  being  known  as  a 

"dot,"  and  a  long  one  as  a  "dash." 
By  a  combination  of  "dots"  and 
"  dashes,"  letters  are  represented  and 
words  are  spelled  out. 


500.  The  Telegraph  System  described 
in  the  preceding  sections  is  known 
as  Morse's,  from  its  inventor.  Fig- 
ure 416  illustrates  diagrammatically 
the  instruments  necessary  for  one 
terminal  station,  together  with  the 
mode  of  connection.  The  arrange- 
ment at  the  other  end  of  the  line  is 
an  exact  duplicate  of  this  one,  the 
two  sections  of  the  battery  being 


Sounder 


•|i|ilh 

Line  Battery 


~~^parth 
Fig.  416 

placed  in  the  line,  so  that  the  negative  pole  of  one  and 
the  positive  of  the  other  are  connected  with  the  earth. 


THE  ELECTRIC  TELEGRAPH 


387 


At  intermediate  stations  the  relay  and  the  local  circuit 
are  connected  with  the  line  in  the  same  manner  as  at  a 
terminal  station. 

501.  The  Electric  Bell  (Fig.  417)  is  used  for  sending  sig- 
nals as  distinguished  from  messages.  Besides  the  gong, 
it  contains  an  electromagnet,  having 
one  terminal  connected  directly  with  a 
binding-post,  and  the  other  through 
a  light  spring  attached  to  the  arma- 
ture (shown  on  the  left  of  the  figure) 
and  a  contact  screw,  with  another 
binding-post.  One  end  of  the  arma- 
ture is  supported  by  a  stout  spring, 
or  on  pivots,  and  the  other  carries  the 
bent  arm  and  hammer  to  strike  the 
bell.  Included  in  the  circuit  are  a 
battery  and  a  push-button  B,  shown 
with  the  top  unscrewed  in  Fig.  418.  Fig.  417 

When  the  spring  E  is  brought  into  contact  with  D  by 
pushing  (7,  the  circuit  is  closed,  the  elec- 
tromagnet attracts  the  armature,  and  the 
hammer  strikes  the  gong.  The  move- 
ment of  the  armature  opens  the  circuit 
by  breaking  contact  between  the  spring 
and  the  point  of  the  screw  ;  the  arma- 
ture is  then  released,  the  retractile  spring 
at  the  bottom  carries  it  back,  and  con- 
tact is  again  established  between  the 
spring  and  the  screw.  The  whole  opera- 
tion is  repeated  automatically  as  long 
as  the  circuit  is  kept  closed  at  the  push-button.  A 
"  buzzer  "  is  an  electric  bell  without  the  hammer  and  gong. 


Fig.  418 


388 


DYNAMO-ELECTRIC  MACHINERY 


V.  THE  TELEPHONE 

502.  The  Telephone  (Fig.  419)  consists  of  a  permanent 
magnet  0,  one  end  of  which  is  surrounded  by  a  coil  of 
many  turns  of  fine  copper  wire  5,  whose  ends 
are  connected  with  the  binding-posts  £  and  t. 
At  right  angles  to  the  magnet,  and  nob  quite 
touching  the  pole  -within  the  coil  is  an 
elastic  diaphragm  or  disk  a  of  soft  sheet- 
iron,  kept  in  place  by  the  conical  mouthpiece 
d.  If  the  instrument  is  placed  in  an  elec- 
tric circuit  when  the  current  is  unsteady, 
or  alternating  in  direction,  the  magnetic 
field  due  to  the  helix,  when  combined  with 
that  due  to  the  magnet,  alters  intermittently 
the  number  of  lines  of  force  which  branch 


Fig.  419 


out  from  the  pole,  thus  varying  the  attraction  of  the  mag- 
net for  the  disk.  The  result  is  that  the  disk  vibrates  in 
exact  keeping  with  the  changes  in  the  current. 

503.  The  Microphone  is  a  device  for  varying  an  electric 
current  by  means  of  a  variable  resistance  in  the  circuit. 
One  of  its  simplest  forms  is  shown  in  Fig.  420.  It  consists 
of  a  rod  of  gas-carbon  A,  whose 
tapering  ends  rest  loosely  in  coni- 
cal depressions  made  in  blocks  of 
.the  same  material  attached  to  a 
sounding  board.  These  blocks 
are  placed  in  circuit  with  a  bat- 
tery and  a  telephone.  While  the 
current  is  passing,  the  least 
motion  of  the  sounding  board,  caused  either  by  sound 
waves  or  by  any  other  means,  such  as  the  ticking  of  a 


Fig.  420 


THE  TELEPHONE 


389 


watch,  moves  the  loose  carbon  pencil  and  varies  the  pres- 
sure between  its  ends  and  the  supporting  bars.  A  slight 
increase  of  pressure  between  two  conductors  resting  loosely 
one  on  the  other  lessens  the  resistance  of  the  contact,  and 
conversely.  Hence,  the  vibrations  of  the  sounding  board 
cause  variations  in  the  pressure  at  the  points  of  contact 
of  the  carbons,  and  consequently  make  corresponding 
fluctuations  in  the  current  and  vibrations  of  the  telephone 
disk. 

504.  The  Solid  Back  Transmitter.  —  The  varying  resist- 
ance of  carbon  under  varying  pressure  makes  it  a  valuable 
material  for  use  in  telephone  transmitters.    Instead  of  the 
loose    contact    of   the   microphone, 
carbon  in  granules  between  carbon 
plates  is  now  commonly  employed. 

The  form  of  transmitter  exten- 
sively used  for  long  distance  work 
is  the  "solid  back"  transmitter 
(Fig.  421).  The  figure  shows  only 
the  essential  parts  in  section,  minor 
details  being  omitted.  M  is  the 
mouthpiece,  and  F  and  C  the  front 
and  back  parts  of  the  metal  case. 
The  aluminum  diaphragm  D  is  held  around  its  edge  by  a 
soft  rubber  ring.  The  metal  block  W  has  a  recess  in  front 
to  receive  the  carbon  electrodes  A  and  B.  Between  them 
are  the  carbon  granules.  The  block  JS  is  attached  to  the 
diaphragm  and  is  insulated  from  W  except  through  the 
carbon  granules.  The  transmitter  is  placed  in  circuit  by 
the  wires  connected  to  IF  and  E. 

Provision  is  made  for  an  elastic  motion  of  the  diaphragm 
and  the  block  E.     Sound  waves  striking  the  diaphragm 


Fig.  421 


390  DYNAMO-ELECTRIC  MACHINERY 

cause  a  varying  pressure  between  the  plates  and  the  car- 
bon granules.  This  varying  pressure  varies  the  resist- 
ance offered  by  the  granules  and  so  varies  the  current. 
The  transmitter  is  in  circuit  in  the  line  with  the  primary 
of  a  small  induction  coil,  the  secondary  being  in  a  local 
circuit  with  the  telephone  receiver.  The  induced  currents 
in  the  secondary  have  all  the  peculiarities  of  the  primary 
current;  and  when  they  pass  through  a  receiver,  it  re- 
sponds and  reproduces  sound  waves  similar  to  those  which 
disturb  the  disk  of  the  transmitter. 

VI.  WIRELESS  TELEGRAPHY 

505.  Oscillatory  Discharges.  —  The  discharge  of  any  con- 
denser through  a  circuit  of  low  resistance  is  oscillatory. 
The  first  rush  of  the  discharge  surges  beyond  the  condi- 
tion   of    equilibrium,    and 
the  condenser  is  charged  in 
the  opposite  sense.     A  re- 
verse discharge  follows,  and 
so  on,  each  successive  pulse 
being  weaker  than  the  pre- 
ceding, until  after    a   few 
surges    the    oscillations 

P.     422  cease.       Figure    422    was 

made  from  a  photograph  of 
the  oscillatory  discharge  of  a  condenser  by  means  of  a  very 
small  mirror,  which  reflected  a  beam  of  light  on  a  falling 
sensitized  plate.  Such  alternating  surges  of  high  frequency 
are  called  electric  oscillations.  Joseph  Henry  discovered 
long  ago  that  the  discharge  of  a  Leyden  jar  is  oscillatory. 

506.  Electric  Waves.— In   1887-1888   Hertz   made   the 
discovery  that  electric  oscillations  give  rise   to  electric 


Heinrich  Rudolf  Hertz  (1857-1894)  was  born  in  Hamburg, 
and  was  educated  for  a  civil  engineer.  Having  decided  to  aban- 
don his  profession,  he  went  to  Berlin  and  studied  under  Helm- 
holtz,  and  later  became  his  assistant.  In  1885  he  was  appointed 
professor  of  physics  at  the  Technical  High  School  at  Karlsruhe, 
and  while  there  he  discovered  the  electromagnetic  waves  pre- 
dicted by  Maxwell,  who  in  the  middle  of  the  century  had  ad- 
vanced the  idea  that  waves  of  light  are  electromagnetic  in  char- 
acter In  1889  he  was  elected  professor  of  physics  at  Bonn, 
where  he  died  at  the  age  of  thirty-seven.  Electromagnetic  waves 
are  called  Hertzian  waves  in  his  honor. 


WIRELESS   TELEGRAPHY 


391 


waves  in  the  ether,  which  appear  to  be  the  same  as  waves 
of  light,  except  that  they  are  very  much  longer,  or  of 
lower  frequency.  They  are  capable  of  reflection,  refrac- 
tion, and  polarization  the  same  as  light. 

Evidence  of  these  waves  may  be  readily  obtained  by 
setting  up  an  induction  coil,  with  two  sheets  of  tin-foil  on 
glass,  Q  and  $',  connected  with  the  terminals  of  the  sec- 
ondary coil,  and  with  two  discharge  balls,  F  and  F',  as 
shown  in  Fig.  423.  So  simple  a  device  as  a  large  picture 


Fig.  423 

frame  with  a  conducting  gilt  border  may  be  used  to  detect 
waves  from  the  tin-foil  sheets.  If  the  frame  has  shrunken 
so  as  to  leave  narrow  gaps  in  the  miter  at  the  corners, 
minute  sparks  may  be  seen  in  a  dark  room  breaking  across 
these  gaps  when  the  induction  coil  produces  vivid  sparks 
between  the  polished  balls,  .Fand  F' .  The  plane  of  the 
frame  should  be  held  parallel  with  the  sheets  of  tin-foil. 
The  passage  of  electric  waves  through  a  conducting  circuit 
produces  electric  oscillations  in  it,  and  these  oscillations 
cause  electric  surges  across  a  minute  air  gap. 

507.  The  Coherer. — A  very  sensitive  device  for  the 
detection  of  electric  waves  is  the  coherer  (Fig.  424). 
When  metallic  filings  are  placed  loosely  between  solid 


392 


DYNAMO-ELECTRIC  MACHINERY 


Fig.  424 


electrodes  in  a  glass  tube  they  offer  a  high  resistance  to 
the  passage  of  an  electric  current ;  but  when  electric  oscil- 
lations are  produced  in  the  neighborhood  of  the  tube,  the 

resistance  of  the  filings  falls  to 
so  small  a  value  that  a  single 
voltaic  cell  sends  through  them 
a  current  strong  enough  to 
work  a  relay  (§  497).  If  the 
tube  is  slightly  jarred,  the  filings  resume  their  state  of 
high  resistance.  A  slight  discharge  from  the  cover  of  an 
electrophorus  (§  398)  through  the  filings  lowers  the  re- 
sistance just  as  electric  oscillations  do.  It  is  thought  that 
minute  sparks  between  the  filings  partially  weld  them  to- 
gether and  make  them  conducting. 

508.  Wireless  Telegraph  Set.  —  The  connections  and  parts 
of  a  simple  wireless  telegraph  set  are  shown  in  Fig.  425. 
One  ball  of  the  secondary  of  the  induction  coil  is  con- 


*    O 

•  Spark  Gap 


Earth 


f  pi 

Coherer  gJ  Filings 

M 


Local  Battery 


Fig.  425 


nected  to  the  earth  and  the  other  to  a  vertical  wire  mast 
for  transmitting  electric  waves.  At  the  receiving  station 
a  similar  wire  mast  is  connected  to  one  end  of  a  coherer 
which  is  joined  in  series  with  a  battery  and  a  relay.  The 


WIRELESS   TELEGRAPHY  393 

make  and  break  circuit  of  the  relay  includes  a  sounder 
operated  by  another  battery.  When  the  transmitter  and 
the  receiver  are  both  properly  adjusted,  the  closing  of  the 
key  at  the  sending  station  produces  sparks  at  the  spark 
gap  of  the  secondary  coil ;  the  electric  waves  sent  out  by 
the  vertical  mast  are  received  by  the  corresponding  one  at 
the  receiving  station,  the  coherer  has  its  resistance  lowered 
so  that  the  battery  works  the  relay,  arid  the  sounder 
responds.  The  sounder  is  usually  made  to  tap  the  coherer 
for  the  purpose  of  restoring  its  sensitiveness  for  the  recep- 
tion of  the  next  waves. 

Wireless  messages  are  transmitted  further  over  the 
ocean  than  over  the  land.  In  fact  the  most  conspicuous 
application  of  wireless  telegraphy  is  between  ships  at  sea, 
or  between  them  and  shore  stations.  It  has  proved  of 
inestimable  value  in  the  timely  assistance  it'  has  brought 
to  disabled  or  sinking  vessels.  Ocean  liners  are  in  daily 
communication  with  other  vessels  or  with  maritime  sta- 
tions. Wireless  systems  of  communication  are  now  a  part 
of  the  equipment  of  naval  vessels.  They  may  thus  be 
kept  in  touch  with  one  another  and  with  the  navy  depart- 
ment of  the  governments  they  represent. 


APPENDIX 


I.    GEOMETRICAL  CONSTRUCTIONS 

The  principal  instruments  required  for  the  accurate  con- 
struction 'of  diagrams  on  paper  are  the  compasses  and  the 
ruler.  For  the  construction  of  angles  of  any  definite  size  the 

protractor  (Fig. 
426)  can  be  used. 
There  are,  how- 
ever, a  number  of 
angles,  as  90°,  60°, 
and  those  which 
can  be  obtained 
from  these  by  bi- 
secting them  and 
combining  their 

parts,  that  can  be  constructed  by  the  compasses  and  ruler  alone. 
A  convenient  instrument  for  the  rapid  construction  of  the 
angles  90°,  60°,  and  30°,  is 
a  triangle  made  of  wood, 
horn,  hard  rubber,  or  card- 
board, whose  angles  are 
these  respectively.  Such 
a  triangle  may  be  easily 
made  from  a  postal  card 
as  follows :  Lay  off  on  the 
short  side  of  the  card  (Fig. 
427)  a  distance  a  little  less 
than  the  width,  as  AB.  Separate  the  points  of  the  compasses 
a  distance  equal  to  twice  this  distance.  Place  one  point  of  the 
compasses  at  J5,  and  draw  an  arc  cutting  the  adjacent  side  at  C. 

394 


Fig.  427 


GEOMETRICAL   CONSTRUCTIONS 


395 


Cut  the  card  into  two  parts  along  the  straight  line  BC.  The 
part  ABC  will  be  a  right-angled  triangle,  having  the  longest 
side  twice  as  long  as  the  shortest  side,  with  the  larger  acute 
angle  60°  and  the 
smaller  30°.  With 
this  triangle  and  a 
straight  edge  the  ^, 

majority  of  the  con- 
structions required 
in  elementary  phys- 
ics can  be  made. 


.  —  To  con- 
struct an  angle  of  90°. 

Let  A  be  the  ver- 


D 


A 
Fig.  428 


E 


tex  of  the  required 
angle  (Fig.  428). 
Through  A  draw  the  straight  line  BC.  Measure  off  AD,  any 
convenient  distance  ;  also  make  AE  =  AD.  With  a  pair  of 
compasses,  using  D  as  a  center,  and  a  radius  longer  than  AD, 

draw  the  arc  mn  ;  with  E  as  a  cen- 
ter and  the  same  radius,  draw  the 
arc  rs,  intersecting  mn  at  F.  Join 
A  and  F.  The  angles  at  A  are  right 

angles. 
D—         "/ 


m 


PROS.  2.  —  To  construct  an  angle 
of 60°. 

Let  A  be  the  vertex  of  the  re- 
— —  quired  angle  (Fig.  429),  and  AB  one 
of  the  sides.  On  AB  take  some 
convenient  distance  as  AC.  With 
a  pair  of  compasses,  using  A  as  a  center  and  AC  as  a  radius, 
draw  the  arc  CD.  With  C  as  a  center  and  the  same  radius,  draw 
the  arc  mn,  intersecting  CD  at  E.  Through  A  and  E  draw  the 
straight  line  AE;  this  line  will  make  an  angle  of  60°  with  AB. 


C 
Fig.  429 


396 


APPENDIX 


PROB.  3.  —  To  bisect  an  angle. 

Let  BAG  be  an  angle  that  it  is  required  to  bisect  (Fig.  430). 
Measure  off  on  the  sides  of  the  angle  equal  distances,  AD  and 

AE.  With  D  and  E  as  centers 
and  with  the  same  radius,  draw 
the  arcs  mn  and  rs,  intersecting 
at  F.  Draw  AF.  This  line 
will  bisect  the  angle  BAG. 

PROB.  4.  —  To  make  an  angle 
equal  to  given  angle. 

Let  BAG  be  a  given  angle; 
it  is  required  to  make  a  second 
angle  equal  to  it  (Fig.  431). 
Draw  DE,  one  side  of  the  required  angle.  With  A  as  a  cen- 
ter and  any  convenient  radius,  draw  the  arc  mn  across  the  given 
angle.  With  D  as  a  center  and  the  same  radius,  draw  the 
arc  rs.  With  s  as  a  center  and  a  radius  equal  to  the  chord  of 


C     D 

Fig.  431 


E 


mn,  draw  the  arc  op,  cutting  rs  at  G.  Through  D  and  G 
draw  the  line  DF.  This  line  will  form  with  DE  the  required 
angle,  as  FDE. 

PROB.  5.  —  To  draw   a  line   through   a  point  parallel  to  a 
given  line. 

Let  A  be  the  point  through  which  it  is  required  to  draw 
a  line  parallel   to   BG  (Fig.  432).      Through   A  draw  ED, 


GEOMETRIC  A  L  CONS  TR  UCTIONS 


397 


cutting  BC  at  D.     At  A  make  the  angle  EAG  equal  to  EDO. 
Then  AG  or  FG  is  parallel  to  BC. 


G 


Fig.  432 

PROB.  6.  —  Given  two  adjacent  sides  of  a  parallelogram  to  com- 
plete the  figure. 

Let  AB  and  AC  be  two  adjacent  sides  of  the  parallelogram 
(Fig.  433).     With  C  as  a  center  and  a  radius  equal  to  AB, 

W 
r-. 


Fig.  433 

draw  the  arc  mri.  With  B  as  a  center  and  a  radius  equal  to 
AC,  draw  the  arc  rs,  cutting  mn  at  .D.  Draw  CD  and  BD. 
Then  ABDC  is  the  required  parallelogram. 


398 


APPENDIX 


II.   CONVERSION  TABLES 


1.   LENGTH 


To  reduce 

Multiply  by      To  reduce 

Multiply  by 

Miles  to  km.  . 

1  60935     Kilometers  to  mi.     .     . 

0.62137 

Miles  to  m. 

1609.347         Meters  to  mi.       .     .     . 

0.0006214 

Yards  to  m 

0  91440     Meters  to  yd 

1  09361 

Feet  to  m  

0.30480     Meters  to  ft  

3.28083 

Inches  to  cm. 

2  64000     Centimeters  to  in.    .     . 

0.39370 

Inches  to  mm. 

25.40005     Millimeters  to  in      . 

0.03937 

2.   SURFACE 

To  reduce 

Multiply  by       To  reduce 

Multiply  by 

Sq.  yards  to  m.2      .     . 

.     0.83613      Sq.  meters  to  sq.  yd.  . 

.     1.19599 

Sq.  feet  to  m.2    .     .     . 

0  09290      Sq.  meters  to  sq.  ft.     . 

.  10.76387 

Sq.  inches  to  cm.2    .     . 

.     6.45163      Sq.  centimeters  to  sq.  in 

.     0.15500 

Sq.  inches  to  mm.2 

645.163          Sq.  millimeters  to  sq.  in 

.     0.00155 

3.   VOLUME 

To  reduce 

Multiply  by       To  reduce 

Multiply  by 

Cu.  yards  to  m.3      .     . 

.     0.76456      Cu.  meters  to  cu.  yd.  . 

.     1.30802 

Cu.  feet  to  m  3    .     .     . 

0.02832      Cu  meters  to  cu   ft.    . 

.  35.31661 

Cu.  inches  to  cm.3  .     . 

.  16.38716      Cu.  centimeters  to  cu.  in 

.     0.06102 

Cu.  feet  to  liters      .     . 

.28.31701      Liters  to  cu.  ft.  .     .     . 

.     0.03532 

Cu.  inches  to  liters 

.    0.01639      Liters  to  cu.  in.       .     . 

.  61.02337 

Gallons  to  liters  .     .     . 

.     3.78543      Liters  to  gallons      .     . 

.     0.26417 

Pounds  of  water  to  liters 

.     0.45359      Liters  of  water  to  Ib. 

.     2.20462 

4.   WEIGHT 

To  reduce 

Multiply  by       To  reduce 

Multiply  by 

Tons  to  kgm  

907.18486      Kilograms  to  tons  .     . 

0.001102 

Pounds  to  kgm.    .     .     . 

0.45359      Kilograms  to  Ib.      .     . 

2.20462 

Ounces  to  gin.      .     .     . 

28.34953      Grams  to  oz.       ... 

0.03527    ' 

Grains  to  gm  

0.064799      Grams  to  grains      .     . 

15.43236 

CONVERSION   TABLES 


399 


5.   FORCE,  WORK,  ACTIVITY,  PRESSURE 


To  reduce  Multiply  by 

Lb.-weight  to  dynes,  .  444520.58 
Ft.-lb.  to  kgm.-m.  .  .  .  0.138255 
Ft.-lb.  to  ergs  .  .  .  13549  x  108 
Ft.-lb.  to  joules  .  .  .  1.3549 
Ft.-lb.  per  sec.  to  H.P.  18182  x  10"7 
H.P.  to  watte  ....  745.196 
Lb.  per  sq.  ft.  to  kgin. 

per  m.2 4.8824 

Lb.  per  sq.  in.  to  gin. 

per  cm.2 70.3068 

Calculated  for  g  =  980  cm., 


To  reduce  Multiply  by 

Dynes  to  Ib.-weight,  22496  x  10'10 
Kgm.-m.  to  ft.-lb.  .  .  .  7.233 
Ergs  to  ft.-lb.  .  .  0-7381  x  10'7 
Joules  to  ft.-lb.  .  .  .  0.7381 
H.P.  to  ft.-lb.  per  sec.  .  550 

Watts  to  H.P 0.001342 

Kgm.  per  m.2  to  Ib.  per 

sq- ft 0.2048 

Gm.  per  cm.2  to  Ib.  per 

sq.  in 0.01422 

or  32.15  ft.  per  sec.  per  sec. 


6.   MISCELLANEOUS 


0.01602 
0.004329 

14.69640 


To  reduce  Multiply  by 

Lb.  of  water  to  U.S.  gal.  0. 11983 
Cu.  ft.  to  U.S.  gal.  .  .  7.48052 
Lb.  of  water  to  cu.  ft.  at 

4°  C.      .     

Cu.  in.  to  U.S.  gal.     .     . 
Atmospheres   to  Ib.    per 

sq.  in 

Atmospheres  to  gm.  per 

cm.2 1033.296 

Lb.-degrees  F.  to  calories,  252 

Calories  to  joules  .  .  .  4.18936 
Miles  per  hour  to  ft.  per 

sec 1.46667 

Miles  per  hour  to  cm.  per 

sec.    ,  44.704 


To  reduce  Multiply  by 

U.S.  gal.  to  Ib.  of  water,  8.345 

U.S.  gal.  to  cu.  ft.       .     .  0.13368 

Cu.  ft.  of  water  at  4°  C. 

tolb 62.425 

U.S.  gal.  to  cu.  in.      .     .  231 

Lb.  per  sq.  in.  to  atmos- 
pheres    0.06737 

Gm.  per  cm.2  to  atmos- 
pheres    0.000968 

Calories  to  Ib. -degrees  F.  0.003968 

Joules  to  calories   .     .     .  0.2387 

Ft.  per  sec.  to  miles  per 

hour 0.68182 

Cm.  per  sec.  to  miles  per 

hour  0.02237 


400 


APPENDIX 


III.   MENSURATION  RULES 


Area  of  triangle 

Area  of  triangle 

Area  of  parallelogram         : 

Area  of  trapezoid  : 

Circumference  of  circle      : 

Diameter  of  circle  : 

Area  of  circle  : 

Area  of  ellipse  : 

Area  of  regular  polygon 
Lateral  surface  of  cylinder 
Volume  of  cylinder  : 

Surface  of  sphere  : 

Volume  of  sphere 
Surface  of  pyramid  ^ 
Surface  of  cone        / 
Volume  of  cone 


=  £  (base  x  altitude). 


—  a)(s—b)  (s  —  c)  where  s= 
base  x  altitude. 

Altitude  x  \  sum  of  parallel  sides. 
diameter    x  3.1416. 
f  circumference  •*•  3.1416. 
1  circumference  x  0.3183. 
t  diameter  squared  x  0.7854. 
I  radius  squared  x  3.1416. 
product  of  diameters  x  0.7854. 
|  (sum  of  sides  x  apothem). 
circumference  of  base  x  altitude. 
=  area  of  base  x  altitude. 
t  diameter  x  circumference. 
14  X  3.1416  x  square  of  radius. 
(  diameter  cubed  x  0.5236. 
"if  Of  radius  cubed  x  3.1416. 
=  |  (circumference  of  base  x  slant  height). 

=  \  (area  of  base  x  altitude). 


Lord  Rayleigh  (John  William  Strutt)  was  born  at  Essex  in 
1842,  and  graduated  from  Cambridge  University  in  1865.  In 
1 884  he  was  appointed  professor  of  experimental  physics  in  that 
institution,  and  three  years  later  he  was  elected  professor  of  natu- 
ral philosophy  at  the  Royal  Institution  of  Great  Britain.  His  work 
is  remarkable  for  its  extreme  accuracy.  The  discovery  of  argon 
in  the  atmosphere,  while  attempting  to  determine  the  density  of 
nitrogen,  was  the  result  of  a  very  minute  difference  between  the 
result  obtained  by  using  nitrogen  from  the  air  and  that  from 
another  source.  Nearly  every  department  of  physics  has  been 
enriched  by  his  genius.  His  treatise  on  Sound  is  one  of  the  finest 
pieces  of  scientific  writing  ever  produced.  His  determination  of 
the  electrochemical  equivalent  of  silver  and  the  electromotive 
force  of  the  Clark  standard  cell  are  contributions  of  the  first  im- 
portance to  modern  electrical  measurements. 


TABLE  OF  DENSITIES 


401 


IV.  TABLE  OF  DENSITIES 

The  following  tables  gives  the  mass  in  grams  of  1  cm.3  of  the  sub- 
stance :  — 


Agate 2.615 

Air,  at  0°  C.  and  76  cm. 

pressure 0.00129 

Alcohol,  ethyl,  90%,  20°  C.  0.818 

Alcohol,  methyl  ....  0.814 

Alurn,  common    ....  1.724 

Aluminum,  wrought     .     .  2.670 

Antimony,  cast   ....  6.720 

Beeswax 0.964 

Bismuth,  cast      .    -.     .     .  9.822 

Brass,  cast 8.400 

Brass,  hard  drawn    .     .     .  8.700 

Carbon,  gas 1.89 

Carbon  disulphide    .     .     .  1.293 

Charcoal 1.6 

Coal,  anthracite  .     .1.26  to  1.800 

Coal,  bituminous     .   1.27  to  1.423 

Copper,  cast 8.830 

Copper,  sheet 8.878 

Cork 0.14  to  0.24 

Diamond 3.530 

Ebony 1.187 

Emery 3.900 

Ether 0.736 

Galena 7.580 

German  silver      ....  8.432 

Glass,  crown 2.520 

Glass,  flint      ...    3.0  to  3.600 

Glass,  plate 2.760 

Glycerin 1.260 

Gold 19.360 

Granite 2.650 

Graphite 2.500 

Gypsum,  crys 2.310 


Human  body    ....  0.890 
Hydrogen,  at  0°  C.  and 

76  cm.  pressure      .     .  0.0000896 

Ice 0.917 

Iceland  spar     ....  2.723 

India  rubber     ....  0.930 

Iron,  white  cast    .     .     .  7.655 

Iron,  wrought  ....  7.698 

Ivory 1.820 

Lead,  cast   .....  11.360 

Magnesium       ....  1.750 

Marble 2.720 

Mercury,  at  0°  C.       .     .  13.596 

Mercury,  at  20°  C.     .     .  13.558 

Milk 1.032 

Nitrogen,  at  0°  C.  and 

76  cm.  pressure      .     .  0.001255 

Oil,  olive 0.915 

Oxygen,  at  0°  C.  and  76 

cm.  pressure      .     .     .  0.00143 
Paraffin  .     .     .     0.824  to  0.940 

Platinum 21.531 

Potassium 0.835 

Silver,  wrought     .     .     .  10.56 

Sodium 0.970 

Steel 7.816 

Sulphuric  Acid      .     .     .  1.84 

Sulphur 2.033 

Sugar,  cane      ....  1.593 

Tin,  cast 7.290 

Water,  at  0°  C.     .     .     .  0.999 

Water,  at  20°  C.   .     .     .  0.998 

Water,  sea .....  1.027 

Zinc,  cast 7.000 


402 


APPENDIX 


M 


V.     GEOMETRICAL  CONSTRUCTION  FOR  REFRACTION 
OF  LIGHT 

The  path  of  a  ray  of  light  in  passing  from  one  medium  into 
another  of  different  optical  density  is  easily  constructed  geomet- 
rically. The  following  problems  will  make  the  process  clear : 

First. — A  ray  from  air  into  water.  —  Let  MN  (Fig.  434)  be 
the  surface  separating  air  from  water,  AB  the  incident  ray  at 
JB,  and  BE  the  normal.  With  B  as  a  center  and  a  radius  BA 

draw  the  arcs  mn  and  Cs.  With 
the  same  center  and  a  radius  f  of 
AB,  (-f  being  the  index  for  air  to 
water),  draw  the  arc  Dr.  Produce 
AB  till  it  cuts  the  inner  arc  at  D. 
Through  D  draw  DC  parallel  to 
the  normal  EF,  cutting  the  outer 
arc  at  C.  Draw  BC.  This  will 
be  the  refracted  ray,  because 

— —  =  -,  the  index  of  refraction. 

\j  Jj  O 

When  the  ray  passes  from  a 
medium  into  one  of  less  optical 
density,  then  the  ray  is  produced 
until  it  cuts  the  outer  or  arc  of 
larger  radius,  and  a  line  is  drawn  through  this  point  parallel 
to  the  normal.  The  intersection  of  this  line  with  the  inner  arc 
gives  a  point  in  the  refracted  ray  which  together  with  the 
point  of  incidence  locates  the  ray. 

If  the  incident  angle  is  such  that  this  line  drawn  parallel  to 
the  normal  does  not  cut  the  inner  arc,  then  the  ray  does  not 
pass  into  the  medium  at  that  point  but  is  totally  reflected  as 
from  a  mirror. 

It  is  immaterial  whether  the  arcs  Dr  and  Cs  are  drawn  in 
the  quadrant  from  which  the  light  proceeds,  or,  as  in  the 
figure,  in  the  quadrant  toward  which  it  is  going. 


Fig.  434 


GEOMETRICAL   CONSTRUCTION 


403 


Second. —  Tracing  a  ray  through  a  lens.  —  Let  MN  represent 
a  lens  whose  centers  of  curvature  are  C  and  C",  and  AB  the 
ray  to  be  traced  through  it  (Figs.  435,  436).  Draw  the  normal, 


OB,  to  the  point  of  incidence.     With  B  as  a  center,  draw  the 
arcs  mn  and  rs,  making  the  ratio  of  their  radii  equal  the  index 
of  refraction,  f .     Through  p,  the  intersection  of  AB  with  rs, 
draw  op  parallel  to   the   normal,  C'B,  and  cutting  mn  at  o. 
Through  o  and  B  draw  oBD ;    this  will  be  the  path  of  the  ray 
through   the   lens. 
At  D  it  will  again 
be    refracted ;     to 
determine     the 
amount,  draw  the 
normal     CD     and 
the    auxiliary   cir- 
cles, xy  and  uv,  as  N 
before.       Through                                  FiS- 436 
the  intersection  of  BD  produced  with  xy,  .draw  It  parallel  to 
the  normal  CD,  cutting  uv  at  /.     Through  D  and  I  draw  DH] 
this  will  be  the  path  of  the  ray  after  emergence. 


INDEX 


[References  are  to  pages.} 


Aberration,        chromatic,        227  ; 

spherical,  200,  217. 
Absolute,    scale    of    temperature, 

252  ;  unit  of  force,  84  ;  zero,  253. 
Absorption  spectra,  230. 
Accelerated  motion,  74. 
Acceleration,  73  ;  centripetal,  79  ; 

of  gravity,  99. 
Achromatic  lens,  227. 
Action  of  points,  301. 
Adhesion,  6  ;  selective,  7. 
Aeroplane,  278. 
Agonic  line,  292. 

Air,  compressibility   of,  57  ;  com- 
pressor,   60 ;    pressure    of,    50 ; 

weight  of,  49. 
Air  brake,  69. 
Air  columns,  laws  of,  172. 
Air  pump,  61  ;  experiments  with, 

62. 

Airships,  64. 
Alternator,  378. 
Altitude  by  barometer,  55. 
Ammeter,  349. 
Ampere,  334. 
Amplitude,  113. 
Analysis  of  light,  226. 
Aneroid  barometer,  54. 
Annealing,  10. 
Anode,  318. 
Antinode,  170. 
Arc,  carbon,  381  ;   inclosed,  382  ; 

open,  382. 
Archimedes,  principle,  41. 


Armature,  346,  371 ;  drum,  375. 
Artesian  well,  38. 
Athermanous  substances,  272. 
Atmosphere,  a  unit  of  pressure,  53. 
Atmospheric  electricity,  313. 
Attraction,         electrical,         293  j 

molecular,  19,  25. 
Aurora,  315. 

Balance,  132. 
Balloons,  64. 
Barometer,  aneroid,  54;  mercurial, 

54  ;  utility  of,  55. 
Baroscope,  64. 
Battery,  storage,  331. 
Beam  of  light,  181. 
Beats,  161. 
Bell,  electric,  387. 
Blind  spot,  225. 
Body,  1. 
Boiling,  263. 
Boiling  point,  effect  of  pressure, 

264  ;  on  thermometer,  245. 
Bottle  imp,  43. 
Boyle's  law,  58;  inexactness  of, 

59. 

Bright  line  spectra,  230. 
Brittleness,  9. 
Buoyancy,  40;  of  air,  63;  center 

of,  42. 

Caloric,  242. 
Calorie,  257. 
Camera,  photographer's,  223. 


405 


406 


INDEX 


Capacity,  dielectric,  306  ;  electro- 
static, 304 ;  thermal,  257. 

Capillarity,  22  ;  laws  of,  22  ;  re- 
lated to  surface  tension,  24. 

Capstan,  134. 

Cartesian  diver,  43. 

Cathode,  318  ;  rays,  365. 

Caustic,  201. 

Cell,  voltaic,  317  ;  chemical  action, 
in,  318. 

Center,  of  buoyancy,  42 ;  of 
gravity,  99 ;  of  oscillation,  114 ; 
of  percussion,  114  ;  of  suspension, 
113. 

Centrifugal  force,  109;  illustra- 
tions of,  110 ;  its  measure,  110. 

Centripetal  force,  109. 

Charge,  residual,  307  ;  seat  of,  307. 

Charles,  law  of,  252. 

Choke  coil,  379. 

Chord,  major,  162  ;  minor,  162. 

Chromatic  aberration,  227. 

Circuit,  closing  and  opening,  318  ; 
divided,  350  ;  electric,  318. 

Circular  motion,  71. 

Clarinet,  171. 

Clinical  thermometer,  247. 

Coherer,  391. 

Cohesion,  6. 

Coil,  choke,  379  ;  induction,  359  ; 
primary,.  357  ;  secondary,  357. 

Cold  by  evaporation,  263. 

Color,  233  ;  complementary,  237  ; 
mixing,  235 ;  of  opaque  bodies, 
234  ;  of  transparent  bodies,  234  ; 
primary,  236. 

Commutator,  372. 

Composition  of  forces,  88 ;  of 
velocities,  91. 

Compressibility  of  gases,  57. 

Concave,  lens,  210  ;  mirror,  193  ; 
focus  of,  194,  213. 


Condenser,  305  ;  action  of,  361. 

Conduction,  of  electricity,  332  ; 
of  heat,  267. 

Conductor,  electrical,  296  ;  charge 
on  outside,  300  ;  magnetic  field 
about,  340. 

Conservation  of  energy,  124. 

Convection,  270. 

Convex,  lens,  210 ;  mirror,  193 ; 
focus  of,  194,  212. 

Cooling,  law  of,  272. 

Corpuscles,  370. 

Coulomb,  302. 

Couple,  87. 

Critical  angle,  208. 

Crookes  tubes,  365. 

Crystallization,  26. 

Current,  electric,  316  ;  convection, 
270  ;  detection  of,  321 ;  heating 
effects  of,  338  ;  induced  by  cur- 
rents, 356  ;  induced  by  magnets, 
354  ;  magnetic  properties  of,  340  ; 
mutual  action  of,  342  ;  strength 
of,  334. 

Curvilinear  motion,  79. 

Cyclonic  storms,  55. 

Daniell  cell,  324. 

Day,  sidereal,  13  ;  solar,  13. 

Declination,  291. 

Density,  44 ;  of  a  liquid,  46 ;  of  a 
solid,  44 ;  bulb,  47  ;  tables  of, 
401. 

Derrick,  134. 

Deviation,  angle  of,  205. 

Dew  point,  266. 

Diamagnetic  body,  282. 

Diathermanous  body,  272. 

Diatonic  scale,  163. 

Dielectric,  305  ;  capacity,  306  ;  in- 
fluence of,  306. 

Diffraction,  240. 


INDEX 


407 


Diffusion,  16,  18,  19. 
Dimensions,  3  ;   measurement  of, 

11. 

Dipping  needle,  291. 
Discharge,  oscillatory,  314,  390. 
Dispersion,  226. 
Drum  armature,  375. 
Dry  cell,  326. 
Dryness,  266. 
Ductility,  9. 
Dynamo,    371  ;     compound,    375 ; 

series,  375  ;  shunt,  375. 
Dyne,  84. 

Earth,  a  magnet,  290. 

Ebullition,  263. 

Echo,  153. 

Efficiency,  128. 

Effusion,  17. 

Elasticity,  28  ;  limit  of,  28 ;  of 
form,  28  ;  of  volume,  28. 

Electric,  bell,  387  ;  circuit,  318 ; 
current,  316  ;  current  detection, 
321  ;  motor,  376  ;  telegraph,  384  ; 
waves,  390. 

Electrical,  attraction,  293  ;  distri- 
bution, 300  ;  field,  298  ;  ma- 
chines, 309 ;  potential,  302 ; 
repulsion,  293  ;  resistance,  332  ; 
wind,  301. 

Electrification,  293  ;  atmospheric, 
313;  by  induction,  298;  kinds 
of,  294  ;  nature  of,  297  ;  simulta- 
neous development  of  both  kinds 
of,  296 ;  unit  of,  302. 

Electrode,  318. 

Electrolysis,  327  ;  laws  of,  329 ; 
of  copper  sulphate,  328;  of 
sodium  sulphate,  327  ;  of  water, 
329. 

Electrolyte,  317,  318. 

Electromagnet,  344. 


Electromotive  force,  318,  335  ;  in- 
duced by  magnets,  354  ;  induced 
by  currents,  356. 

Electron  theory  of  matter,  369. 

Electrophorus,  308. 

Electroplating,  330. 

Electroscope,  295. 

Electrostatic,  capacity,  304 ;  in- 
duction, 298. 

Electrostatics,  293. 

Electrotyping,  330. 

Energy,  1,  120 ;  conservation  of, 
124  ;  dissipation  of,  124  ;  kinetic, 
120  ;  measure  of,  122  ;  potential, 
120  ;  transformation  of,  123. 

English  system  of  measurement,  11. 

Equilibrant,  87. 

Equilibrium,  87 ;  kinds  of,  101 ; 
molecular,  27  ;  of  floating  bodies, 
42. 

Erg,  118. 

Ether,  177. 

Evaporation,  cold  by,  263. 

Expansion,  coefficient  of,  251 ; 
force  of.  254  ;  of  gases,  57r  250  ; 
of  liquids,  250  ;  of  solids,  248. 

Extension,  3-. 

Eye,  224. 

Falling  bodies,  104  ;  laws  of,  106. 
Field,    electrical,   298 ;    magnetic, 

287. 

Field  magnet,  371,  374. 
Floating  bodies,  42. 
Fluids,  31  ;   characteristics  of,  31 ; 

pressure  in,  31. 
Fluorescope,  368. 
Flute,  171. 
Focus,  194  ;   conjugate,  196,  213  ; 

of  lens,  212  ;  of  mirrors,  194. 
Foot,  5. 
Foot  pound,  118. 


408 


INDEX 


Force,  83 ;  composition  of,  86 ; 
graphic  representation  of,  85  ; 
how  measured,  84  ;  lines  of,  292  ; 
molecular,  19 ;  moment  of,  130  ; 
of  expansion  and  contraction, 
254  ;  parallelogram  of,  88  ;  reso- 
lution of,  89 ;  units  of  83. 

Force  pump,  68. 

Forced  vibrations,  155. 

Fountain,  39;  vacuum,  63. 

Fraunhofer  lines,  231. 

Freezing  point,  245;  mixtures,  262. 

Friction,  127;  uses  of,  128. 

Fundamental,  tone,  168;  units,  13. 

Fusion,  260;  heat  of,  261. 

Gallon,  12. 

Galvanometer,  d1  Arson  val,  347. 

Galvanoscope,  321. 

Gas  engine,  276. 

Gases,  32;   compressibility  of,  32; 

expansion    of,    250 ;    media    for 

sound,  149 ;  thermal  conductivity 

of,  268. 

Gassiot's  cascade,  363. 
Gauge,  water,  38. 
Geissler  tube,  364. 
Geometrical    constructions,    394, 

402. 

Grain,  13. 
Gram,  12. 
Gramme  ring,  373. 
Graphic  methods  in  sound,  174. 
Gravitation,  98  ;  law  of,  100. 
Gravitational  unit  of  force,  83. 
Gravity,   99;    acceleration  of,   99; 

cell,  325;  center  of,  99;  direction 

of,  99;  specific,  44,  48. 

Hardness,  9. 

Harmonic,  curve,  145;  motion,  71, 
80. 


Harmonics,  170. 

Heat,  242;  conduction  of,  267;  con- 
vection of,  270;  due  to  electric 
current,  338;  from  mechanical 
action,  274;  kinetic  theory  of, 
242;  lost  in  solution,  262;  me- 
chanical equivalent  of,  274; 
measurement  of,  257 ;  nature  of, 
242;  of  fusion,  261;  of  vaporiza- 
tion, 265,  radiant,  271;  related 
to  work,  274;  specific,  258; 
transmission  of,  267. 

Helix,  342;  polarity  of,  342. 

Holtz  machine,  309. 

Hooke's  law,  29. 

Horizontal  line  or  plane,  99. 

Horse  power,  120. 

Hydraulic,  elevator,  34;  press,  33. 

Hydrometer,  47. 

Hydrostatic  paradox,  37. 

Hypothesis,  2. 

Images,  by  lenses,  214;  by  mirrors, 

189,  199;  by  small  openings,  183. 
Impenetrability,  4. 
Impulse,  94. 
Incandescent  lamp,  383. 
Inclination.  290. 
Inclined   plane,    137;    mechanical 

advantage  of,  137. 
Index  of  refraction,  204. 
Induced  magnetism,  284. 
Induction,  charging  by,  300;  coil, 

359;  electrostatic,  298;  magnetic, 

285 ;  self-,  358. 
Inertia,  4. 

Influence  machine,  309. 
Insulator,  297. 
Intensity,  of  illumination,  184;  of 

sound,  159. 
Interference,     of     light,    238;     of 

sound,  160. 


INDEX 


409 


Intervals,  162;    of  diatonic  scale, 

163;  of  tempered  scale,  165. 
Ions,  318. 
Isobars,  56. 
Isoclinic  lines,  291. 
Isogonic  lines,  292. 

Joule,  119. 

Joule's  equivalent,  274. 

Kaleidoscope,  193. 
Keynote,  163. 
Kilogram,  12. 
Kilogram  meter,  118. 
Kinetic    energy,     120;     measure- 
ment of,  122. 

Kinetic  theory,  18;  of  heat,  242. 
Kundt's  dust  figures,  177. 

Lamp,  arc,  382;  incandescent, 
383. 

Lantern,  projection,  222. 

Law,  Boyle's,  58;  Lenz's  357; 
Ohm's  336;  of  Charles,  252;  of 
cooling,  272;  of  electromagnetic 
induction,  355;  of  electrostatic 
action,  295;  of  falling  bodies, 
106;  of  gravitation,  100;  of  heat 
radiation,  271;  of  magnetic  ac- 
tion, 284;  of  machines,  127; 
Pascal's,  32;  physical,  2. 

Laws,  of  motion,  93;  of  air  col- 
umns, 172;  of  strings,  167. 

Leclanch.6  cell,  326. 

Length,  11. 

Lens,  210;  achromatic,  228;  focus 
of,  212;  images  by,  214. 

Lenz's  law,  357. 

Lever,  130  ;  mechanical  advantage 
of,  132. 

Leyden  jar,  306  ;  theory  of,  307 ; 
charging  and  discharging,  306. 


Lift  pump,  67. 

Light,  179;  analysis  of,  226; 
propagation  of,  181 ;  reflection 
of,  187;  refraction  of,  203; 
speed  of,  180 ;  synthesis  of,  227. 

Lightning,  313  ;  rod,  313. 

Lines,  agonic,  292  ;  isoclinic,  291 ; 
of  magnetic  force,  292. 

Liquefaction,  260. 

Liquid,  2,  32 ;  cohesion  in,  7 ; 
compressibility  of,  32 ;  density 
pf,  46  ;  downward  pressure,  35  ; 
expansion  of,  250  ;  in  connected 
vessels,  38  ;  medium  for  sound, 
149  ;  surface  level  in,  38  ;  surface 
tension  in,  21 ;  thermal  con- 
ductivity of,  268 ;  velocity  of 
sound  in,  161. 

Liter,  11. 

Local  action,  322. 

Lodestone,  281. 

Longitudinal  vibrations,  144. 

Loudness  of  sound,  159. 

Machine,  126 ;  efficiency  of,  128  ; 

electrical,    311;    law    of,     127; 

mechanical  advantage  of,    129 ; 

simple,  129. 

Magdeburg  hemispheres,  63. 
Magnet,  artificial,  282;  bar,  282; 

electro-,    344;    horseshoe,    282; 

natural,  281. 
Magnetic,  action,  284  ;  axis,  283  ; 

field,  287  ;  lines  of  force,   287  ; 

meridian,     283;    needle,      283; 

polarity,    282  ;    substance,    282 ; 

transparency,  283. 
Magnetism,  induced,  284  ;  nature 

of,   286 ;    permanent    and    tem- 
porary,   285  ;    terrestrial,    290 ; 

theory  of,  286. 
Magnets,  281. 


410 


INDEX 


Major  chord,  162. 

Malleability,  9. 

Manometric  flame,  175. 

Mass,  6  ;  units  of,  12. 

Matter,  1  j  properties  of,  3 ;  states 

of,  2. 

Mechanical  advantage,  129. 
Mechanical    equivalent    of    heat, 

274. 
Mechanics,  of  fluids,  31  ;  of  solids, 

83. 
Melting     point,     260 ;     effect    of 

pressure,  261. 
Mensuration  rules,  400. 
Meter,  11. 
Metric  system,  11. 
Micrometer  caliper,  140. 
Microphone,  388. 
Microscope,       compound,       228  ; 

simple,  219. 
Minor  chord,  162. 
Mirror,  189  ;  focus  of,  194  ;  images 

by,     189,      197  ;    plane,      189 ; 

spherical,  193. 
Mobility,  31. 
Moistness,  266. 
Molecular,  forces,  19,  25  ;  motion, 

17  ;  physics,  16. 
Moment,  of  a  force,  130. 
Momentum,  93. 
Monorail  car,  5. 
Motion,      71;     accelerated,      74; 

curvilinear,    71,    79 ;    harmonic, 

71,  80  ;  laws  of,  94  ;  molecular, 

17  ;  pendular,  64  ;  periodic,  89  ; 

rectilinear,      71  ;     rotary,      71 ; 

uniform,  72  ;  vibratory,  80. 
Motor,  electric,  376. 
Musical,  scales,  162  •  sounds,  157. 

Octave,  162. 
Ohm,  333. 


Ohm's  law,  336. 

Opaque  bodies,  179. 

Opera  glass,  222. 

Optical,  center,  211  ;  instruments, 

219  ;  constructions,  402. 
Organ  pipe,  171. 
Oscillation,  center  of,  114  ;  electric, 

390. 

Ounce,  13. 
Overtones,  170,  174. 

Partial  tones,  170. 

Pascal,  experiments,  52  ;  law,  32. 

Pendular  motion,  112. 

Pendulum,  applications  of,  115  ; 
laws  of,  113;  seconds,  115; 
simple,  111. 

Percussion,  center  of,  114. 

Period  of  vibration,  113. 

Periodic  motion,  8. 

Permeability,  289. 

Phenomenon,  1. 

Photographer's  camera,  223. 

Photometer,  185. 

Photometry,  184. 

Physical,  law,  2 ;  measurements, 
10. 

Physics,  1. 

Pigments,  238. 

Pitch,  158;  limits  of,  165;  meas- 
urement of,  158 ;  relation  to 
wave  length,  158  ;  of  screw,  139. 

Plumb  line,  99. 

Pneumatic  appliances,  66. 

Points,  action  of,  301. 

Polarization,  323. 

Polarity  of  helix,  342. 

Poles  of  a  magnet,  283. 

Porosity,  7. 

Potential,  difference  of,  302  ; 
energy,  121 ;  loss  of,  350 ;  unit 
difference  of,  303  ;  zero,  304. 


INDEX 


411 


Pound,  12. 

Power,  119. 

Pressure,  of  fluids,  31  ;  at  a  point 
in  fluids,  36 ;  air,  58 ;  down- 
ward, 35 ;  effect  on  boiling 
point,  264 ;  effect  on  melting 
point,'264  ;  independent  of  shape 
of  vessel,  36  ;  rules  for  comput- 
ing, 37. 

Principle  of  Archimedes,  41. 

Prism,  206 ;  angle  of  deviation, 
207. 

Proof  plane,  296. 

Properties  of  mattery  3. 

Pulley,  135  ;  mechanical  advantage 
of,  136  ;  systems  of,  135. 

Pump,  air,  61  ;  compression,  60 ; 
force,  68  ;  lift,  67. 

Quality  of  sounds,  159 ;  due  to 
overtones,  159. 

Radiation,  271  ;  laws  of,  271. 

Radioactivity,  369. 

Rainbow,  228. 

Rays  of  light,  181. 

Reflection,  diffused,  188  ;  law  of, 
187 ;  multiple,  192  ;  of  light, 
187 ;  of  sound,  153 ;  regular, 
187  ;  total,  208. 

Refraction,  cause  of,  2£1  ;  atmos- 
pheric, 207  ;  laws  of,  206. 

Regelation,  261. 

Relay,  385. 

Resistance,  129  ;  electrical,  332  ; 
laws  of,  333  ;  unit  of,  333. 

Resolution,  of  a  force,  89  ;  of  a 
velocity,  91. 

Resonance,  155,  156. 

Resonator,  HelrnLoltz's,  157. 

Resultant,  86. 

Riveting  hammer,  70. 


Roentgen  rays,  366. 

Scale,  absolute,  252;  diatonic, 
163;  tempered,  164. 

Screw,  138  ;  mechanical  advantage 
of,  139. 

Second,  13. 

Secondary  or  storage  battery, 
331. 

Seconds  pendulum,  115. 

Self-induction,  358. 

Shadows,  182. 

Sidereal  day,  13. 

Sight,  225. 

Singing  flame,  161. 

Siphon,  66  ;  intermittent,  67. 

Solar  day,  13. 

Solenoid,  342  ;  polarity  of,  342. 

Solids,  2  ;  .density  of,  44 ;  ex- 
pansion of,  248  ;  thermal  con- 
ductivity of,  267 ;  velocity  of 
sound  in,  152. 

Solution,  25  ;  saturated,  26  ;  heat 
lost  in,  262. 

Sonometer,  167. 

Sound,  143,  148  ;  air  as  a  medium, 
149 ;  liquids  as  media,  149 ; 
loudness  of,  157  ;  musical,  157  ; 
quality,  of,  159 ;  reflection  of, 
153 ;  sources  of,  148 ;  trans- 
mission of,  149  ;  velocity  of, 
150  ;  waves,  150. 

Sounder,  telegraph,  385. 

Specific  gravity,  44,  48 ;  bottle, 
46. 

Specific  heat,  258. 

Spectroscope,  232. 

Spectrum,  solar,  226 ;  kinds  of, 
229. 

Spherical  aberration,  200,  217. 

Spheroidal  state,  263. 

Spherometer,  140. 


412 


INDEX 


Stability,  102  ;  angular,  43. 

Stable  equilibrium,  101. 

Steam  engine,  275. 

Steelyard,  132. 

Storage  cell,  331. 

Strain,  96. 

Strength   of  an  electric    current, 

334  ;  methods  of  varying,  336. 
Stress,  96, 

Strings,  laws  of,  167. 
Sublimation,  263. 
Submarine  boat,  44. 
Substance,  1. 
Surface  tension,  21 ;  illustrations 

of,  21. 

Suspension,  center  of,  113. 
Sympathetic  vibrations,  155. 
Synthesis  of  light,  227. 

Tables,  conversion,  398  ;  density, 

401. 
Telegraph,     electric,     384 ;     key, 

384  ;  signals,  386  ;  system,  386  ; 

wireless,  390. 
Telephone,  388. 
Telescope,      astronomical,      221  ; 

Galileo's,  222. 
Temperature,     242  ;      measuring, 

243. 

Tempered  scale,  164. 
Tempering,  10,  165. 
Tenacity,  8. 
Theory,  2. 

Thermal  capacity,  257 . 
Thermometer,  244 ;  clinical,  247  ; 

limitations  of,  247  ;  scales,  245. 
Thunder,  313. 
Time,  13. 
Tone,  fundamental,  168;    partial, 

170. 

Torricellian  experiment,  52. 
Transformer,  378. 


Translucent  bodies,  179. 
Transmission  of  heat,  267. 
Transmitter,  389. 
Transparent  bodies,  179. 
Transverse  vibrations,  143. 
Trombone,  171. 
Tuning  fork,  148. 

Units,  10;  of  heat,  257;  of  length, 
11;  of  mass,  12;  of  time  13. 

Vacuum,  Torricellian,  52. 
Vaporization,  262;  heat  of,  265. 
Velocity,  71;  •  composition   of,  91; 

constant,   72;  of  light,    180;    of 

molecules,    18;    of   sound,    150 ; 

resolution  of,  91;    uniform,   72; 

variable,  72. 
Ventral  segments,  170. 
Vertical  line,  99. 

Vibration,  amplitude  of,  113 ;  clas- 
sified, 143;  complete,  113,  143; 

forced,    155;    longitudinal,    144; 

of  strings,   167;    period  of,   113; 

single,    113;    sympathetic,    155; 

transverse,   143;   rate  measured, 

195. 

Viscosity,  31. 
Volt,  335. 
Voltaic  cell,  316;   chemical  action 

in,  318.  . 
Voltameter,  335. 
Voltmeter,  348. 
Voss  electrical  machine,  309. 

Water,  gauge,  38;  waves,  147. 

Watt,  120. 

Wave  motion,  144. 

Waves,  144;  longitudinal,  146; 
electric,  390;  length,  147;  sound, 
150 ;  transverse,  144 ;  water,  147, 

Wedge,  138. 


INDEX 


413 


Weight,  6;    law  of,    101;    of  air, 

49. 

Weston  normal  cell,  335. 
Wheatstone's  bridge,  351. 
Wheel  and  axle,  133;  mechanical 

advantage  of,  133. 
Whispering  gallery,  154. 


Wireless  telegraph,  390. 
Work,  47;   units  of,   118;    useful, 
128;  wasteful,  128. 

X-rays,  367. 
Yard,  12. 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
This  book  is  DUE  on  the  last  date  stamped  below. 

Fine  saddle:  2 5' cents 

fourth  day  overdue** 


14  1947 


APR   27  1948 

140ct'498C 


REC'D  LD 

APR  14 1957. 


BP  16  1957     S 


REC'3  LD 

I3EC3-1959 

25JUL'64Gf 


7Jan'53SS  P 
*»5    1953  LU 


LD 


7J957 

100ct'58BB 


; 


REC'D  LD 

1  7  1959 


LD21-100m-12,'46(A2012s   6)4120 


17Dec'59WW 


4J6 


rf?c 


C  Z  • 


' 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


